CIHM 

ICIVIH 

Microfiche 

Collection  de 

Series 

microfiches 

(i\/lonographs) 

(monographies) 

Canadian  Institute  for  Historical  Microreproductions  /  Institut  Canadian  de  microreproductions  historiques 


iOO>l 


Technical  and  Bibliographic  Notes  /  Notes  techniques  et  bibliograph:ques 


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24  X 


28  X 


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22X 


The  copy  filmed  hare  has  been  reproduced  thank* 
to  the  generosity  of: 

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University  Laval, 
Qu6bec,  Quebec. 

The  images  appearing  here  are  the  best  quality 
possible  considering  the  condition  and  legibility 
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filming  contract  specifications. 


L'exemplaire  film^  fut  reproduit  gr§ce  d  la 
gAnirositA  de: 

Bibliothdque  gin^rale. 
University  Laval, 
Quebec,  Quebec. 

Les  images  suivarites  ont  6x6  reproduites  avec  le 
plus  grand  soin,  compte  tenu  de  la  condition  et 
de  la  nettetA  de  I'exempiaire  filmA,  at  en 
conformity  avec  les  conditions  du  contrat  de 
fiimage. 


Original  copies  in  printed  paper  covers  are  filmed 
beginning  with  the  front  cover  and  ending  on 
the  last  page  with  a  printed  or  illustrated  impree- 
sion.  or  the  back  cover  when  appropriate.  All 
other  original  copies  are  filmed  beginning  on  the 
first  page  with  a  printed  or  illustrated  impres- 
sion, end  ending  on  the  last  page  with  a  printed 
or  illustrated  impression. 


Les  exemplaires  originaux  dont  la  couverture  en 
pepier  est  ImprimAe  sont  fiimis  en  commenpant 
par  le  premier  plat  et  en  terminant  soit  par  la 
derniAre  page  qui  comporte  une  empreinte 
d'impression  ou  d'illustration,  soit  par  le  second 
plat,  salon  le  cas.  Tous  les  autres  exemplaires 
originaux  sont  filmte  en  commenpant  par  la 
premiere  page  qui  comporte  une  empreinte 
d'impression  ou  d'illustration  et  en  terminant  par 
la  derniire  page  qui  comporte  une  telle 
empreinte. 


The  last  recorded  frame  on  each  microfiche 
shall  contain  the  symbol  ^»-  (meaning  "CON- 
TINUED ").  or  the  symbol  V  (meaning  "END"). 
whichever  epplies. 


Un  dee  symboles  suivants  apparaitra  sur  la 
derniire  image  de  cheque  microfiche,  selon  le 
cas:  le  symbols  — »>  signifie  "A  SUIVRE",  le 
symbols  V  signifie  "FIN". 


Maps,  plates,  charts,  etc.,  may  be  filmed  at 
different  reduction  retios.  Those  too  large  to  be 
entirely  included  in  one  exposure  are  filmed 
beginning  in  the  upper  left  hand  corner,  left  to 
right  and  top  to  bottom,  as  many  frames  as 
required.  The  following  diagrams  illustrate  the 
method: 


Les  cartes,  planches,  tableaux,  etc..  peuvent  dtre 
film6s  A  des  taux  de  reduction  diff^rents. 
Lorsque  le  document  est  trop  grand  pour  dtre 
reproduit  en  un  seul  cliche,  il  est  film6  d  partir 
de  I'angle  sup^rieur  gauche,  de  gauche  i  droite, 
et  de  haut  en  bas,  en  prenant  le  nombre 
d'images  n^cessaire.  lies  diagrammes  suivants 
illustrent  la  mithode. 


1 

2 

3 

1 

2 

3 

4 

5 

6 

MICROCOPY    RESOLUTION    TEST   CHART 

(ANSI  and  ISO  TEST  CHART  No.  2) 


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1653   East   Main   Street 

Rochester.   New   York         14609       USA 

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(716)   288  -  5989  -  Fox 


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I.KViv.u,    IMPKOVf         .i.VU   SlMVLII-IKl., 

Dv    THOMAS    ATKINSON.    M.  A,, 

^*^*""'>I.iB   OF  COr.f.   OU.  COLL.,   CAJtBHlDSR. 


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.iLG^EBRA. 


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Bytiie  L^tV; 

/,  ^  RET.  B.  BRIDajj^kD,  P.R.8., 

..  ^J«J*y/J/«^Ao;;...-.'.-c««M^  j-.'lz^.  IUlosoi,h'jimhEa,tLxdla  Coll.,  Ua'tf:>ra 
BEVISfeD,   IH7H0-/3D,  AND  filMPLIFIED  BY 

THOMAS  ATKI&X,  M.A,, 

LaU  ScUlsr  of  Corp.  Ch.  foil.  'ccvibridQe. 


i'nm  t!,c  £au,t  imHn  eMtian,  «.if,  .^bMti.n^  l>i,  ii;e  I3r.ti;cr3  of  r  ^ 


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Kew  York: 

n.  &  J.  SABLIER  &  CO.,  31  BARCLAY  ST. 

Montreal:   275  Notre  Dame  Street. 


1S7C, 


accomp] 

the  man 

^  period 

professei 

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This  s 

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ADVERTISEMENT. 


%' 


% 


The  excellence  of  "Bridge's  Algebra,"  as  an  do- 
,icntary   treatise,    has   long   been  ^  well   known    uid 
extensively  recognised.     In  the.  Preface  to  the  Second 
^Edition  the  author  expressly  stAt^;,  that  "great  pains 
|werc  taken  to  give  to  it  all  the  perspicuitv  and  sinipli- 
|City  which  the  subject  would  a\im  of,  and  to  present 
It  m  a  form  likely  to  engage  th'd- 'attention  ofyoun<. 
f  persons  just  entering  on  their  jiWhematical  studies'' 
ihe  design,  which  he  thus  proposed  to  himself,  was 
accomplished  with  singular  fem^^,~-foT  not  one  of 
the  many  publications  on  Algebra,  which  have  durinc. 
m  period  of  forty  years  issued  from  the  press,  with  the 
professed  object  of  producing  a  more  simple  and  ap- 
propriate introduction  to  the  study  of  the  science  has 
evinced  such  merits  as  justly  ent  J .  it  to  be  placed 
in  comparison  with  the  performance  of  Mr   I3rido-(. 
These  publications  have  accordingly  failed  to  secm-e 
flH'  themselves  the  same  measure  of  public  approba- 

This  small  compendium  embraces  all  which  is  com- 
^'^      ~     "  '""--•'  ''■^'i/(^  ana  cu-jjcnsive  editions,  that 


^- 


Iy 


ADVEimSEMEXT. 


ether  practicallv  useful  or  theoreticn;iy  valuable 
By  .ntroducng  Eyuation,,  and  Problems  at      e  e  "" 
.    .est  stage  po.ible,  a  nove    and  in.s.r  X  V  fure 
»n.ch  the  eCtor  is  ,,ersuaded  eannot  fail  to  ex    to tu 
ounos,tyand  stimulate  the  ardour  of  the  yom?,W 
l-raist,  so  as  to  induee  him  to  pursue  his  studio  t  f. t 
more  than  usual  alacrity,  intelligence,  and  tc  ol- 
nas  been  g,ven  to  the  work.     A  great  ;ariety  of"! 

tSdt  r'    ""';=T  P™!':^'"^.  which  are'^not  Z 

amed  m  former  edit^ors,  have  thus  been  interspersed 

the  several  chapters.;;  Besides  these  addition'm^ 

t.  m.ty  of  arrangement,  or  of  rcnderin<.  ,he  subiee. 

boShatlis';Ir'^''''-''*">""'  ""•^■''  -'  -""-' 
bope   hat  h,s    ilc,n,„M-y  iVeatise  on  Algebra'  would 

h"d  Its  wa.y  ,nto  <^;P>,Uk  Schooh;  where    it  vvas 
.very  well  known,  tWs  branch  of  eduction  was     hi 
but   little  attended   ts-i'    and  if  ,V   „   ""  ^™H'hen] 
honprl   tl,„t.i,-  ,.'.  "   "°**'  confidently 

Goa»oBD,  December,  3347.  ^'  A. 


y  valuable, 
at  the  ear- 
e  feature — 
'  excite  the 
oiing  alge- 
■udies  with 

success — 
ty  of  new, 

not  con- 
terspersed 
ons  many 
ic  of  uni. 
ic  subject 
^ds.     Mr. 
vithout  a 
'«'  would 
e,  it  \va3 
as  [then] 
nfidently 
ice  of  its 
to  many 
^ntlj  at- 
ing  tills 
n. 

T.  A. 


CONTENTS 


t*aAPTKR  r    DEPrN-mONS.  P*®* 


•^HAP.    J  I, 


HnAi, 


On   the  Addition,  Subtraction,  Muitiph^tior; '/uij  'l,-;'.       " 
sion  of  Algebraic  Quantities....  , 

Addition *' 

Simple  Equations '\\ ^ 

Ou  the  Solution  of  Simple'  Equation"  ' .1 

iVoblems "-' 

Subtraction .. .  .!!.'!'.".'*'.'.* ^*"^'^ 

On  the  Solution  of  Sirn*ple*EquatioM .*.' 11 

Multiplication ^^ 

On  the  Solution  of  SimplVEqnVuons;  *.'.'. ]! 

Iroblorns. -° 

Division. _  _  -9-35 

HI.  On  Algebraic  Fractions .  ^  "^  ^ 

\^LtT7^  "^"^  'M^iipiicati;,;:  ;;.;,■  d^:  ^ ' 

sion  ot  fractions 

On  the  Solution  of  Sim^e  Equaiions'. ." ^ 

Problems f55 

On  tlie  Solution  'oV  Siniple  'E^uat.c 

more  unknown  Quantities.. 


61-Ct} 

.ons,  containing  two  or 


Probl 


ems. 


60 


'^a-vr.  \\.  On  Involution  and  Evolution ^^'"^^ 

'^^.^:^,^ ^-be. anj-sim;^;':^,;^;  '' 

On  the  Involution  of  Comnnnn^  * \  I. !„V_'- L'  V.'  "H '^^ 

oa  the  Evoiuto  of  Aig.srQu^s;::.^!!!!!':?:;;  ,'j 


Tl  CONTENTS. 

PA«t 

Oil  the  Investigation  of  the  Rule  for  the  Extraction  of  the 
Square  Root  of  Numbers 81 

Chap,    V      Oii  Quadratic  Equations. . . , 83 

On  the  Solution  of  pui-e  Quadratic  Equations. . , , 81 

On  the  Solution  of  adfected  Quadi'atic  Equations 85 

Problems 

On  the  Solution  of  Problems  producing  Quadi-atic  Equa- 
tions  .*.  ...92-93 

On  the  Solution  of  Quadratic  Equations  containing  two 
unknown  Quantities 98 

Ciup  VI.     O41  Arithmct'c,  Geomotric,  and  Harmonic  Progression...   10"2 

Problems 105-109 

On  Geometric  Progression 10*» 

On  tlie  Summation  of  an  infinite  Series  of  Fractions  m 
Geoniotiic  Pi'ogression ;  and  on  the  method  of  finding 

the  value  of  Cu'cuhiting  Decimals 113 

Problem , 116 

On  Harmonic  I'rogression 117 

Chat.  VII.  On  Permutations  and  Combinations. II9 

AJ'i'KNDix.   On  the  Different  Kinds  of  Numbers 125 

On  the  Four  Rules  of  Arithmetic 126 

On  the  Two  Terms  of  a  Fraction 127 

On  Ratios  rdxd  Projx)i'tion3 128 

On  the  Squares  of  Numbers  and  their  Roots. 128 

On  the  Factors  and  Submultiples  of  a  Number 13G 

On  Odd  and  Even  Numbers 131 

On  Progi'essions .......131 

On  Divisible  Numbers  without  a  Remainder. iaS 

Properties  and  various  Explanations. 181 

Mi*cellaueous  Problems. ,   187 

4 


BRIDGE'S  ALGEBRA 


CHAp-pjilR   I. 

.*  *  •  * 

DEFSNKTIONS. 

pressed  by  melns  of  Written  S  o.- Zboh     Th^  "'",  "" 
u^d^todono.  „u.nbo.  o.  ^UUies^a^'te  I^.t:« 

semed  by  the/.,  lette.  ofl^^^^X^^r' 

3.  Unknown  or  undctermlhH  quantities'  aro  „«,  oil  * 
pressed  b,  the  to.  letters  ortlii>,%abet  "as":,  ^X:'' 

4.  1  no  m?//W^5  of  quantities  thnt  ;«  +k^  z        . 
quantities  are  ti  be  i^l^S^&'^X^'Z^TfJr' 

34- sl:  "Th"""  f^^  P'-i"S.  timbers  Tefore'thi/f  Is  "C 
rtear„  tfr  ^  ^"-"'''  »  '^  ^^a^l^^Se^^  .Vu^ 

.«  =  G«.    This  ,vn>bo.  ist„ed  the' tV4«%.'"*''  '" 

rore-&^+pt»:t''ir  taT'i'^^  ^ 

siiinc  th  ntr  as  5  •  and  a  ^  h  a.  1  ,  -"-""^  »^  +  2  is  tiie 

.,  whatever  be  the "fl^e^of  at^'aXr    "  "^  °'  "'  ''  ""' 


titios^r^^i^neltoSEl^Xhai:;^^  ^-?^<'  --^  or^ 

or ././.,./«.«,,/  quantities  ?_3.  llmv  i«  LiL,    f''*"^"''.*^  rei>rcscnt  Aown 

•-6.  WhatKs  the  sign  of  equality  ^-o.  W  In    \TJtror  urt"'+?'^  ' 


^^LGEBRA. 


7.  Tlie  sign  —  (read  minus)  signifies  that  the  quantitv  tc 
which  it  is  prefixed  is  to  be  subtracted.  Thus  3  —  2  is  the 
same  thing  as  1  ;  a  —  b  means  the  difference  of  a  and  b,  or  b 
tal.-en  from  a ;  and  a  +  6  —  .r,  signifies  that  x  is  to  be  sub 
iracted  from  the  sum  of  a  and  b. 

8.  Quantities  which  have  the  sign  +  prefixed  to  them  an* 
called  positive,  and  those  "which  have  the  sign  —  set  before 
them  are  termed  tiegative  quantities.  When  there  is  no  sign 
befoi  e  a  quantity  -\-  is  understood  :  thus  a  stands  fbr  +  a.  '^ 

9.  The  symbol  X  (read  m^oj' is  the  sigii  of  multiplication. 
and  signifies  that  the  quan^itje^  between  Avhich  it  'is  placed 
are  to  be  multiplied  togethcji-;  -  Thus,  6x2  means  that  (J 
IS  to  be  multiplied  by  24,:aud  a  X  b  X  c,  signifies  that 
a,  6,  c,  are  to  be  multiplied  together.  In  the  place  of  this 
symbol  a  dot  or full-point  h  min  used.  Thus,  a.b.c,  means 
the  same  as  (a  X  6  X  c.  T'h©.  product  of  quantities  repre- 
sented by  letters  is  usually  ^fekpressed  by  placing  the  letters 
in  close  contact,  one  after  another,  according  to  the  position 
in  which  they  stand  in  the  alphabet.  Thus,  the  product  of  a 
into  b  is  denoted  by  ab  ;  of  a,, ^  "and  .r,  by  abx ;  and  of  3.  a,  .r, 
and  y,  by  '6axy.  """<> 

10.  In  algebraical  computations  the  word  therefore  often 
occurs.  To  express  this  \vovd  the  symbol  .*.  is  generally 
made  use  of.  Thus  the  sentence  "  therefore  a  -{•  b  is  equal 
to  c  +  G?,"  is  expressed  by  "  .•!  a  +  i  =  c  -f  c/." 

•EXAMPLES. 

Ex.  1 .  In  the  algebraical  expression,  a  4-  b  —  cAet  a  —  Q 
i  =  7,  c  =  3  ;  then 

a  +  b  -  c=    94-7--3 
=  10-3 
=  13 
Ex.  2.  In  the  expression  ax  ■}-  ay  ~  xy,  let  i  —  b  x  —  % 
y  n=  7  J  then,  to  fnid  its  value,  we  have 

ax-\-ay  -xy—    5x2  +  5x7-2x7 
=.  10  +  35  —  14 
^45-14 
=  31 


M :,  ii 


':):*'''^^'  rc:ia  f — o.  Wiuuirt  iiioaiiL  by  wwrni-e  nnd  whal 
oy  mgative  qunnlitiea  i—<i.  Write  down  tlio  don  of  muUipUcation  f  T«  m\ 
other  m«r^  used  to  denote  rnnltiplioation  ?  Wlien  ia  no  Bvinbol  usod?.- 
1 0.  W  1ml  symbol  is  a»cd  to  denote  the  word  therefore  f 


e  quant itv  tc 

3-2  is  the 

a  and  b,  or  b 

is  to  be  sub 

I  to  them  aiv 
—  set  bclurc 
pro  is  no  siijij 
s  tor  -f-  (I- 
\ultipIication, 

it  is  pliioed 
ncans  that  (5 
signifies  that 
place  of  this 
.b.c,  means 
itities  repre- 
ig  the  letters 

the  position 
product  of  a 
ad  of  3,  a,  .r, 

erefore  often 
is  generally 
+  ^  is  equal 


f,  let  a  —  9, 


—  5,  .r  n=  % 
X7 


DEFi.Mi  OXS.  3 

tlio  numerical  values  of  tl.    .^s^^^l^^^^J^  ^    '  '"' 


13. 
3. 

C5. 


(1.)  a  +  b  +  c  +  x.  ^,,, 

(2.)  a-i  +  c-^  +  y.  ^4^^^ 

(3.)  Ob  +  3ac  -  ic  +  4c^  ~  xy.  Ans. 

(4.)  aic  -  abd  +  bed  -  acx.  Ans. 

(5.)  3a4c  +  Aacx  -  SAcfo;  +  a^ry.  Ans.  176. 
11.  ITie  symbol  -r  (read  divided  by)  is  the  sinn  of  .Jh,: 
^^hlch  It  IS  placed,  is  to  be-VJiVided  by  the  latter.     llT" 

exp;esseYr"'''^f-''\  ^^^  ^^^^«'«"  ^«  "^o'e  simp  t' 
expressed   by  makmg   the  ^-brn^er  quantity  the  numerator^ 

^nd  the  latter  the  denominator  pf  a  fraction  :  thus^mc.rs 

a  d^ided  by  ^  and  is  usualJ)^^r  the  sake  of  brevity,  read 

exaa«*i:es. 
Kx.  1.  If  a  =.  2,  i  =  3  ;  fReh,  we  find  the  values  of 


/I  \  ?«  _  3  X  2 


_0. 
15 


i>^       5X3 

(2)  ?l±i.=  ?i<_?_!^''^' 
8a -3/;       8X2  -^Vs 


IG  —  0  "~  7  ~  ^• 


Ex.  2.  If «  =  3,  i  =  2,  c  =  1,  fn,d  the  numerical  values  of 


(1.) 


8r/  4-  c 


-^W.9. 


10 


(2.),-+"'-" 


(3.). 


3«  +  b~bc' 
oJt  -f-  ac—bc 


ir 

Ans.  1. 


.      7 
Ans.--. 

8 


2ab—2ac  +  ic  ' 


of  tnnes,  the  product  is  called  a  now.r  nf  fh.  ..,„..tZ. 

__^ J  "-      - —    • jtutiititV. 


11.  By  wli.it  symbol  is  d......^., 

division  over  expressed  in  any  other 
;«'">?/•  of  u  quantity?  ^ 


ivision  denoted?      W] 

ninnncr?— 12.  Wi 


int  is  its  nnmc 
!i«t  id  menut  i 


7  tli« 


ALGEBRA. 


13.  Pnnmrs  are  usually  denoted  by  placing  above  the  quan. 
tity  to  tiie  right  a  small  figure,  which  indicates  how  often  th€ 
quantity  is  multiplied  into  itself.     Thus, 

a     -     -     -    -     X\iQ  first  power  of  a  is  denoted  by  a  (a'). 
«Xa-     -    -     the  2d  power  or  s5'?/are  of  a     "      a*. 
«XaX«    -     the  3d  power  or  CM^^- of  a         "      a\ 
«  X  a  X  a  X  a  the  4th  power  of  a  «      a\ 

The  small  figures  \  \  \  &c.,  set  over  a,  are  respectively 
called  the  index  or  exjponent  of  the  corresponding  power  of  a. 

14.  The  roots  of  quantities. are  the  quantities  from  which 
the  powers  are  by  successivf5.t«vltiplication  produced.  Thus 
the  root  of  the  square  niim|?<;r46  is  4,  because  4x4=  Ig' 
and  the  root  of  the  cube  num-b^o:  27  is  3,  since  3  X  3  X  3  J 

15.  To  express  the  roods' }^  quantities  tne  symbol  ^/  fa 
corruption  of  r,  the  first  le#f.>in  the  word  radix,)  witti  the 
proper  index,  IS  employed.    Thus,  .    ^ 

W  or  y/a,  expresses  f  he  square  root  of  a. 
/«  "     '•  "     •'•".thec^/ierootofa. 

y«  "     «  "■"■fhe/or«-^/.rootofa. 


■«l 


EXTAMPLES. 

Ex.  1.  If  a  =  3,  6  =  2 ;   then  a«  =3x3  =  9    a^-^^ 
3X3  =  27,4^  =  2X2X2X2  =  16.  •'^  «  - '^  X 

,  ^^x.  2.  If  g  =  64  ;   then   -^/a  =  ^'64  =  8,  \/a  =  V04  = 
^"4  X  4  X  4  =  4,  Va  =  V64  =  2. 


Ex.  3.  In  the  expression 


ax''  +  /;» 


6:^^rrc'^^ta=3,i  =  5,c=:2 
What  is  the  numerical  value  ? 


Here  ««'  +  4'  =  3  X  6  x  G  +  5  x  5  =  108  +  25  =  133 
and  />.r  -  a«  -  c  =  5  X  6  -  3  X  3  -  2  =  30  -  0  --  2  =  1  ii 


ax'^  4-  h 


bx  -—a^  —  c 


133_ 


19 


IK^wJ^rf^^TT*''"'",''*'''^^-^*-  Whntaro  tl.o  roots  of  qunntities  ?- 
"^lilot^^l^SllT ''''''''' ''''''''''''''    ^^P--^^   ^Virat'irtbe 


ovc  the  qiian. 
how  often  the 


respectively 
5  power  of  a. 

s  from  which 

luced.    Thus, 

4X4  =  16, 

X3X3^ 

ymbol  -/,  (a 
ix,)  with  the 

fa. 


9,  a'  =  3  X 
a  =  V04  = 

=  5,  c  =  2 

-  25  =^  133, 

D  -  2  =  1<) 


DEFTXITXOXS. 

£x.  4.  If  a  =  1,  i  =  3,  ,  ,^  5^  ^^  ^^  ^^^  ^j^^^  .^^^j^^^^  ^^ 

(1.)  «=  +  25^.c:.  . 

Ans.    2. 

(2.)  a'  +  3b'  _  c>.  ,         „ 

(3.)  «•  +  24'  +  3c'  +  4d'.  ^„,  9j 

(4.)3a'i-25'c  +  4o'_4a'A  ^„,' ij,; 

■^  3  ^  3  ^  3  •  -4'«.  51. 

Ex  5.  Let  «  =  04,  4  =  81.  0  =  1 :   M  the  values  of ' ' 

'■     ^''+V^"-  An.  n. 

(2.)  v/a+^i+y,.  ^,_^_jg 

(3.)  V^ 

10    When  several  quantities  are  to  bo  taken  as  one  onan 
tj^!,,  they  are  enclosed  in  bracket,,  as    (     )      M    f    1 

semed  tt  +1  '^  ■  ^''  -.  \  ^'S"'«-  *••"  *e  qua.  tity  ri  J. 
=  12.     '^^'  *"''  •■•  (''  +  ''-<^)-(<'-0  =  4x3 

.1.-fltcnVel;ts2nt';rr\;«t%:,+iV;i-'>,t-  -O- 
(1.)  (a+4)  .  (.+rf)  =  (3+2)  .  (3+5)=5i8=4"o 
(2.)  ("+4) <:+rf=(3+2) 3+5=5x3+5=20. 
(3.)  a+6c+,fc3+2x3+5=3+0+5=14 

iectuely.     Thus,  a-i~c  is  the  same  as  a-(b-c) 

ofi^fr^^io^tlT-r:,:.?/.  "---  ™^.de"on,iuat„r 

___  " '•-'  '  "'  ^  ^''^'■'^  oi  viiicuium,  cor. 


llnl^vSr"orn!;rtt  r  ^'"•"•'^^  ^-^^-  ^hnt  ,«  a  t 
^ardcda.a:rue;;r„;;'°  '^"■"•^•••^^-r  "nd  denominator 


i.H  a  vinculum  f    Mny  tlio 
f>t'u  liucUoii  be  re- 


r, 


ALGEBRA. 


responding,  in  fact,  in  Division  to  the  braokct  in  MuUlpVca, 
Hon.     Thus, implies  that  the  whole  quantity  a-f-  b-^  i 


5 


is  to  be  divided  by  5. 

18.  Z^•A•e  quantities  are  such  as  consist  of  the  same,  letter  or 
the  same  combination  of  letters ;  thus,  5«,  and  7«,  4«A  ^nd 
9a/;,  26.c^  and  66.r^  &c.,  are  called  like  quantities ;  and  7//*- 
like  quantities  are  such  as  consist  oi  different  letters,  or  oi' clif 
jerent  combinations  of  letters ;  thus,  4a,  Si,  lax,  5bx'  &o 
are  unlike  quantities.  '  '        » 

10.  Algebraic  quantities  have  also  different  denominations 
•according  to  the  number  of  terms  (connected  by  the  sic^n  + 
or  — )  of  which  they  consist :  thus, 

_  a,  2b,  Sax,  &c.,  quantities  consisting  of  one  term,  are  called 
mpiple  quantities. 

a4-.r,   a   (jtuantity  consisting  of  two    terms,   is   called    a 
fjinomial. 

bx-hy-z,  a  quantity  consisting  of  three  terms,  is  call-d  a 
trinomial. 


CHAPTER    II. 

ON    THE    ADDITION,    SUBTRACTION,    MULTIPLICATION,    AND    DIVI- 
SION,   OF    ALGEBRAIC    QUANTITIES. 

ADDITION. 

20.  Addition  consists  ii.  collecting  quantities  that  are  like 
bUo  one  sum,  and  connecting  by  means  of  their  proper  signs 
those  that  are  unlike.  From  the  division  of  algebraic  numv 
titles  into  positive  and  nef/ative,  like  and  unlike,  there  arise 
Uiree  cases  of  Addition. 

Case  I. 
To  add  like  quantities  with  like  sif/ns. 

21.  In  this  case,  the  rule  is  "To  add  the  coefficienls  of  the 
Bevcral  quantities  together,  and  to  the  result  annex  the  com- 


13.  Wl.at  are  lay  nnrt  what  aro  iinUh  qnntititles  ?_19.  Wliat  is  a  oimnL 
quant. ty?  \\  Unt  hnhho,»lal  and  wl.at  u //v;«.>,;»;«rAo.  fwh^  Si 
tudaion  ot  al-ebra  consist  ?     Iiuo  how  many  casos  is  ic  divided  i 


ADDITION. 


in  MuWiplkcu 
ntity  a-f-i'v— < 

mme  letter.,  or 
7«,  \ah  and 
ic3 ;  and  vn- 
ers,  or  ofdif 
'X,  5b£\  &e.» 

enominations 
Y  the  sign  + 

M,  are  called 

Js   called   a 

I,  is  calkd  8 


AND    DIVI- 


;hat  are  H^-e 

proper  signs 

braic  quan- 

there  arise 


icnts  of  the 
X  the  CO  lu- 


II at  is  fi  simpU 

III  what  dc«« 
led^ 


«ion  sp,  and  the  common  letter  or  l-.t'ors  •"  f .    -f  •        •  i 
from  the  comrnon  principles  of  AH  Im    ic  if  +o  /  '!  ^"^-"I 

be  -153^    '         -^^  '  ^'  "^^^^  ^^g^ther,  their  .mn  must 


Ext  ]. 

S^-f  2a—  5i 
4^-j-  8a— 
9^+  4a- 
5x+  7a  — 


Ex.  2, 


<w. 


7h 
Gb 
9b 

23x+24^^:::su 


Ex.  4. 

3r'+4.r'-  ar 


7^'+  3x1/-  5bc 

}lx'+  5.ry-  4/;c 

:»'+  4^y-      l>c 

__Ji+__^/-.   2bc 

^^£+23xi^m^ 

Ex.  5. 

'7a'-3a'b^2'>b'-~3b' 
4a8-  aV>+  oii_  /,3 

«'-2a''6+3a62__5// 

5a»-3a7>4.4a/;2_oi« 


Ex.  3. 

4a'-  3a-4-  I 
2a'*—  a-'-f]7 
5«'--  2a24-  4 
3a«-  7a*-|-  3 

loa^^— I4a-.f35 


Ex.  6., 

2^V--3^+  2 
4^V-2^+  ) 

3.ry- 5^-1- 10 
xy—  ar-j_]5 


,u,ttrter4S^^,ob--d  tha.  s„,.e  of  .ho 
always  undentood     STZ.A  r  "u""'  "«''•"  «■•  1  " 

Ex.  2,  wo  say,  1 +  1 +  ,1^9+7    fo^P-  "if  >»'  '^"''™»  "f 
^'+5-19;  andsoofthtri       "'  ">"'»""«',  2+1+., 

Case  II. 

To  add  like  qnanfUics  with  unlike  si<;ns. 

mfrlvf''^^  ^''^  compound  quantity  a4.b-c4-d    .    «       • 
positive  or  negative,  according  as  the  turn  oT f f  '      '•.'  '' 
torms  is  greater  or  less  than  thS  sum  of  <?'  ^'"^  ^''''^'^'^'^ 

nggregate  or  sum  of  th.  q  nn  ^tipT  o      ^^  f%^'^^'^'"  ^'"«%  the 
+2_«,^.dof;.,eciuantitieT7^^^^^^^^^  -iii    '- 

tor  yn  <^""   ^"~ ■  "^        «j(y  -\~^o  — j.j,'>*  Will    ii.i   Ai-2 . 

92.  Sento  tl,c  rnio  in  ,!,„  ,,,  „^ 


I 


8 


ALGEBRA. 


Serins  above  the  negative  ones  is  2a,  and  k  the  latter  4/.' 
Hence  this  general  rule  for  the  addition  of  like  quantities  with 
unlike  signs,  "Collect  the  coefficients  oHhe  positive  terms  iuti^ 
one  sum,  and  also  of  the  negative;  subtract  the  Zm^r  of  these 
sums  from  the  ffreater  ;  to  this  difference,  annex  the  sign  of 
the  greater  together  with  the  common  letter  or  letters,  and  the 
result  will  be  the  sum  required." 

If  the  aggregate  of  the  positive  terms  be  equal  to  (hat  of 
the  negative  ones,  then  this  difference  is  equal  to  0;  and  com. 
scquently  the  sum  of  the  quantities  will  be  equal  to  0,  as  in 
the  second  column  of  Ex.  2,  following. 


Ex.  1. 

4ar-  — 3x--|-   4 

^x^—hx-\-    1 

7.i:-^+2x—   4 

~_^4.r+]« 

1  Ix'^— 9.r4-   y 


Ex.  2. 

— 7ai+3^;c—  xy 
—   ab-\-2bc-\-4xi/ 

3a6—  ic+2jry 
—2ab+4:bc^Sxi/ 

5ab  —  Sbc-^  xy 

-~2ab  +3x// 


Ex.  3. 

-fxr'-f  13.r' 
— 2.r'—  4r'^ 


7.r»+ 


.r' 


9.c^- t4.i-^ 
-133:^-2.y'- 

—  4.r*— (),r^ 


E\.  4. 

4/''-  2.r+3// 

—  x^-{-  4x—  y 

7a;'—     x+^y 

^x^+2\x~2y 


Ex.  5. 

5a»— 2ai4-   /^« 

~a'+  ai--2i' 

4a='— 3a<!>-f-  i" 

2a='+4ai-4/'>» 


Ex.6. 

4.iV+2a7/— 3 
■  xhf—  xy—\ 

Sxy-}-4xi/-^5 
■9.ry-2./-y4-9 


Case  III. 

^  23.  Tiiere  now  only  remains  the  case  where  imlike  quan- 
titles  are  to  be  added  togethei,  which  must  be  done  by  col. 
lecting  them  together  into  one  line,  and  annexing  their  proper 
signs  ;  thus,  the  sum  of  3jr,-2a,+56,-4?/,  is  Sx-2a-\-5b-. 
~-'  '    r ■    ■■^'■-■-  «ii-^   z.!:tiftiv  4Ljaiini,ius  are  rnixeu  to. 


23.  State  the  rule  in  tlio  2cl  nnd  8d  casos. 


1  f% 

'•"mm 


the  latter  4^)' 
quantities  with 
itive  terms  into 
3  lesser  of  these 
!X  the  sign  of 
letters,  and  the 

fual  to  fhat  of 
to  0;  and  f.'(>;i. 
Lial  to  0,  as  in 


Ex.  3. 


—   4'Z-().r^ 


Ex.  6. 

U'y4-2A7/-3 

XY—    X1J—I 


•2.;'jy+9 


unlike  quan- 
!  done  by  col. 
J  their  proper 

re  mixou  to- 


SIMPLE   EQUATIOXS.  q 

gether,  as  in  the  foJlowin<r  eximnles  whor-n  ^u 

maybe  simplified  by  colfeothrC '.,    '"^"^^    ''^  expressions 

will  coalesce  intolesum^^^^^^^      ^"'^'''  '"^^^^  'Quantities  aa 


Ex.  1. 

^ab  -\.    X  —  y 
4c     -  o^  4.  ^ 

r)a6  —  3c  -f.  t/ 
jy    +  a;^  -  2y 


Collecting  together  ///l-e^  quan. 
titles,  and  beginmng  with  'Sub 
\vehave3aH-5a6  =  8a6;  -f- «' 

i  2,7-"^  ^•'' 7"^-^^+ Ay 

f.y y;  4c  -  3c  =4- p. 

besides  which  there  are  the  two 


quantities  +  c/  and  +  x'^  whlrh  X       .        ,  ^^''"  ''^^^  ^^e  two 


Ex.2. 

4.r-2xy+l  --%+4a:' 
4y   4-3:i:^-y2_^a:y--   «« 

3£--^_  144.2//+ 12^'2:;. 


Here  4.r'~a;==3jr 

^2xij-\-xy=i^xi/ 
+  l--15=:-i4 
-3y+4y+y=^_Oy 
+4ar'+3.2:H5a:-'=-flbpS 

-2.r=-2x. 


SIMPLE  EQUATIONS. 

.?'m//on.  Equations  ntlT/'^^r  ^^.P^^««^«n  ^^  called  a„ 
problems,  con\"st  o  "quantir  S^:fr^^V^'  ^«^"^^-"  ^^ 
others  nnknozvn.  iZtxTilTl'yf  ""^''^  ^'"  '^'^^^^^^  ^^^ 
:^  is  an  ^//iX^not..^,.  ^z^an//^,,  anrlTi^.  tni  "  ^"  «^q"ation  in  which 
^vill  make  2^+3  L^r+T  eoulun  "t  '\'"'^  a  number  as 
which  here  saiisJi!^letlSt  '^t  "'^f*  ^'^^  """^^^^^ 
ijl^stl,  4,  since  ix^+'^lTCanr  '^T^^^' 

the  unknown  ouamitv  \eW,„h\„^  ■    J-  ■'™  ™luo  o( 

by  inspection^  ,"  3^n.'"':'l,!'J!'.  "I  '.'"^  »«'nple  been  found 


;*S 


10 


ALGEBRA. 


'-]").  l!.  efTccting  tlie  solution,  the  several  stops  of  the  pro- 
ooss  nuist  be  conducted  by  means  of  the  following  axioms, 
and  in  strict  accordance  with  them  : — 

(1.)  Things  which  are  equal  to  the  same  thing  are  equal  to 
one  another. 

(2.)  If  equals  be  added  to  the  same  or  to  equals,  the  sums 
will  be  equal. 

(3.)  If  equals  be  subtracted  from  the  same  or  from  equals 
the  remainders  will  be  equal. 

(4.)  If  equals  be  multiplied  by  the  same  or  by  equals,  the 
products  will  be  equal. 

(5.)  If  equals  be  divided  by  the  same  or  by  equals,  th*' 
Quotients  will  be  equal. 

(6.)  If  equals  be  raised  to  the  same  power,  the  powers  will 
be  equal. 

(7.)  If  the  same  roots  of  equals  be  extracted,  the  roots 
will  be  equal. 

These  axioms,  exclusive  of  the  first,  may  be  f/eneralized, 
and  all  included  in  one  very  important  prmci pie ^  w'hich  should' 
in  every  investigation  in  which  equations  are  concerned  be 
carefully  borne  in  mind  ;  viz.,  that  whatever  is  done  to  one  side 
of  an  equation  the  same  thing  must  be  done  to  the  other  side, 
in  order  to  keep  up  the  equality. 

_  26.  If  an  equation  contain  no  power  of  the  unknown  quan- 
tities higher  than  the,^;-*-^,  or  those  quantities  in  their  simplest 
form,  it  is  called  a  Simple  Equation. 


ON  THE  SOLUTION   OF  SIMPLE  EQUATIONS   CONTAINING  ONLY    ONS 

UNKNOWN    QUANTITY. 

^  27.  The  rules  which  are  absolutely  necessary  for  the  solu- 
tion of  simple  aquations,  containing  only  one  unknown  quan- 
tity,  may  be  reduced  to  four,  each  of  which  will  in  its  proper 
place  be  formally  enunciated  raid  exemplitied. 


25.  What  arc  the  axioms  omployed  in  tlio  solution  of  conations  nnd 
Btave  the  ireneral  principio  wliicli  is  based  upon  tlicni  ?— 2(j.  What  id  a'bini- 
pie  equation  i 


I 


SIMPLE   EQUATIOXa 


eps  of  tho  pro- 
lowing  axioms, 

ig  are  equal  to 

jiials,  the  Slims 

)r  from  equals, 

by  equals,  the 

by  equals,  tht» 

;he  powers  will 

;tcd,  the  roots 

3e  f/eneralised, 
',  which  should 
concerned  be 
'one  to  one  side 
'  the  other  side, 

mknown  quan- 
.  their  simplest 

HNG  ONLY    ONE 

y  for  the  solu. 
mknown  quan- 
1  in  its  propcr 


11 


Rule  I. 

-^qual."  ^    ^  '''"^^'  ^^^.  q»«tients  arising  will  be 

T^x.  I.    Let  2..=  14;    then   d;vldin<;  both   sides    of   the 
equation   by  2,  we  have    -=y.   hnt   ^^  i   1^*     ^ 

•  •  -t- —  4 ,  y 

Ex.2.    Let  az=b  +  c;   then  ^=*-±f.    w  ''"^     . 

Ex.3.  Let  x-\-2x-\-4x-j-Qx=52. 
Collecting  the  terms,  18a-=52. 
Dividing  both  sides  of  the  equation  by  KS, 

x=4. 
Ex.  4.  Let  Gx~~4z-{-Sx-z=SQ. 
The  terms  being  added  as  in  Case  2d  of  Addition, 

4.r=36. 
Dividing  each  side  of  the  equation  by  4, 


Ex.  5    10.^=150. 


Ex.  6.  Sz-^4x-h7.v=:84. 
Ex,  7.  8jr-5.r-|-4x--2x=25. 
Ex.8.  ^2x-3x^4x~-x==24. 


Ans.  ar=15. 
Ans.  z~  (). 
Ans.  x=  r> 
Ans.  x=i  0. 


f  eqiifirions,  nncj 
.  \\  hat  is  11  ttiui- 


the  suLjoincd  cvaranles  h,  1 1,  ?^    ,    •''  "'"^  "  '"■"  ^'«  ^<'™  I" 

'" "  -""ci  :  as  jor  Histi 


i^ 


ALGEBRA. 


(.V,)  Dut  the    ost  of  5  lbs.  by  the  .juestion=30*. 

M.)  TFicrefore,  the  price  of  1  ]b.  in  shiHings  X5=305. 

r/nilt.  '^'''^''''  "•'  ^'"^'^'"^  ^^  ^'  ^'"  ^^*^'"  tJ^«  F'*^ 

Thy  ...v-ral  steps  of  this  soluti.  a  expressed  a].rebr-,If...!iu 
Hould  take  the  following  more  compendious  fonn-"^ 
(] .)  Let  a;=the  price  of  1  lb.  in  shillings. 
(2.)  ITien  5.r=the  price  of  1  lb.  in  shillingsx  5 
(3.)  But  the  cost  of  5  lbs.  is  by  the  questbn^GO,?. 
(4.)  .-.bxzzzSOs. 

Jvl"'"^  ''■"^'^''  ^^''^  ^^  '^'  P™^  «^  l^b.,  as  was  re- 

It  will  be  seen  by  steps  (2)  and  (3)  of  this  example  that 

here  are  two  dist.nct  expressions  for  the  same  tl^S^'  and 

hat  m  step  (4)  these  expressions  are  made  equal  fe' each 

other.     In  Iraming  equations  from  problems,  this  wil    in  all 

.ases  take  place.     As  a  second  exaSiple  let  this  p'blem  be 

A  house  and  an  orchard  are  let  for£2S  a  year,  but  the  rent 
oHhe  house  is  6  tmies  that  of  the  orchard,  -"l^'d  the  rent  "f 

The  rent  of  the  house  is  equal  to  that  of  G  orchards-  we 
maj^  therefore  change  the  house  into  G  orchards,  and  w^shlu 

Tnkn.u  the  sum  of  the  rents  of  the  orchard,  we  get 

7  times  the  rent  of  the  orchard=£2S 
and  the  7th  part  of  each  side  of  the  equation  being  t^kvi^ 

The  rent  of  the  orcnard=£4  • 
ui:i  /.  the  rent  of  the  house=6  times  the  rent'of  the  orciiard 

=6  times  £4 

INow  to  i-ve  to  these  operations  an  alffebraioal  shani> 
i-;et    -z^i'ie  rent  oj  the  orchard  in  £  *   ' 

then  8..  .  ''        «     j^ouse      « 


hiMPLE   EQL'ATIOXS 


as  was  re- 


But  by  the  condition  of  the 
Hent  of  the  orchard     0 


question 
times  ri 


o/^he  orchard— £28. 


or,       7x=£2b. 
«iHl,  dividing  each  side  of  the  equation  by  7, 
x=£4,  the  rent  of  the  orchard 
and  ...  Gx=Gx£4=£-24,  the  rent  of  the  house 

-  ..h^.r.,  .hat  'onr-pa.^^^;  tells  r  ^^  "^^ 
h>    9.       A   person   unacquainted  with  alcreb.a   mllu    ^Z 

.;.^^rcat_dimcu,„.  s„,vi  this  quosLfi.'-L'"'!!;.?..;,;;^ 

(2.)  The  greater  part  .nust  exceed  the  less  by  9 
(ii.)  But  It  IS  evident  that  the  greater  and  less  n^-ts  ,,i  i  i 
iogother  must  be  equal  to  the  whole  number  35    ^        ^"'' 

Jii^^f^  s^i^:^  ""''^  "-^  '<=-  P-'  with  the 

(T.)  Hence,  twice  the  less  part  is  equal  to  26 
2,^'iZX  af  ?;:!olf' ^'^'  """  *^  ^'''^x"-'  -  «9ual  to 
oJi^i^:^^^, '"»  ^™'-  P-t  -eeds  the  .... 

Lut  hy  adopting  the  method  o/ algebraic  wo/^/Zn,,    ft     r.r 

(1.)  Let  the /f-ss  part      -  -  .     __ 

(•^.)  Then  the  greater  part =a-4-9 

(3.)  But  greater  part+1 


part 


=3; 


""■' lum-mt^j^y^^i^^T-^.- 'Ill   II iilniKwii  J 


U 


ALGEBRA. 


iliii 


l!ll 


(4.)  ,.-.  a;+9-4-.r 

(5.)  or2.r-j-9  .... 

(6.)  .'.2x 

(7.)  or2a;    .     .    . 

(8.)  .'.X  {less  part) 

(9.)  and  x-{-  9  {r/reafer  part) 


=35, 

-  =35. 

.     =35-9. 

=26. 

20 
=^=13. 

-  =13+9=22. 


29.  Hjivmg  tlnis  explained  the  manner  in  which  the  several 
5|-eps  ni  the  solution  of  an  arithmetical  question  may  be  ex- 
pressed  HI  the  language  of  Algebra,  we  now  Droceed  to  its  ex- 
emplification. 

PROBLEMS. 

PROB.  1.  A  dessert  basket  contains  30  apples  and  pears 
but  4  timps  as  many  pears  as  apples.  How  many  are  there 
01  each  sort  ?  "^ 

Lot  a;=the  number  of  apples ; 
then,  as  there  are  4  times  as  many  pears  as  apples, 
4.r= the  number  of  pears. 
But  by  the  question  the  apples  and  pears  to(rether=30 

.-.  a,-+4^=30. 
.■idding  the  terms  containing  .r, 

5.r=30. 
Dividing  each  side  of  this  equation  by  5, 

x=G,  the  number  of  apples, 
.'.  the  number  of  pears =4.r =4x0 =24. 

TRon.  2  In  a  mixture  of  10  lbs.  of  black  and  green  tea, 
there  was  3  times  as  much  black  as  green.  Find  the  quaiititv 
Ot  each  sort.  ^  -^ 

Let  .t=the  number  of  lbs.  of  green  tea 
thcn3x=  "         «  «  |,]j^ck^      ' 

But  the  black  tea -f  the  green  tea  =  10  lbs. 
^„      .  .'.^r 4-3,^  =  10  lbs. 

Collecting  the  terms  which  contain  x, 

„.  .^.  ,  4.r  =  101bs. 

.r=r4  lbs.  of  green  tea, 
.*.  the  black  tea  -3i'=3x4=  12  lbs. 


s 

I 

,1- 

t 


SIMPLE  EQUATIONS.  15 

Prob.  3.  An  cqunl  mixture  of  black  tea  at  5  ihUWucr^  ■,  iK 
«nd  ot  green  at  7  shilllnffs  a  lb  costs  4  .rnhnn!  jj^""  '^  "^• 
lbs.  were  there  of  each  sort  ?  ^      '''''     ^^"'^'  "'""J' 

Let  _a:=the  number  of  lbs.  of  each  sort  • 
then  o.r= the  cost  of  the  black  in  shillings 
and  7x=         «         «  j^         ,,      »  ' 

But  cost  of  biack+the  cost  of  green=4  guineas=:84. 

.-.  5a:+7.r=:84 
12jr=84 

.'.a; =7  lbs.  of  each  sort. 
Prob.  4.  The  area  of  the  rectangular  floor  of  a  school-rpom 
|s^^^hO  square  yards,  and  its  breadth  0  yards.     ^r^H 

a;  X  9= the  area  of  the  floor 
.'.  9.r=180  ' 

and  .-.    ar=:20  yards,  the  length  required. 

Prob.  5.  Divide  a  rod  15  feet  long  into  two  n-irts  .<.  fl.  ,t 
the  one  part  may  be  4  times  the  length  of  the  other.'  ' 

Let  ar  =  the  less  part,  1         ,  15t> 

then  4a:=the  greater  part,  '~r' "~ir 1 

i>ow  <hese  two  parts  make  together  15  ft 
.•.a:+4x—15ft. 

5.r:=:I5ft. 

,  ^,  .'.x=:  3  ft.  the  less  part 

and  the  greater  part=4  times  3  ft.  ==12  ft. 

Prob.  G.  A  horse  and  a  saddle  weie  boucrht  for  £.10  bnt 

Ajis.  £30  and  £4. 

^'In^.  Itichard's  share  =.-18, 
-■l;*.?.  Andrew's  share  =  (}' 

8aiM7f^hnf  ^^  ^''"°  ""'^^^  ^"^^^  "^'^"3^  '"^^'W^'s  he  had 
saiJ,  It  1  had  twice  as  manv  mm-o  T  .u^Ja  u..,._  o^  "V^''i^J. 

many  had  he 'I  *  '  '  ■"-'"'■'^  "^*^'^  '•^"'     iiow 

J/w.  12, 

Prob.  9.  A  bookseller  sold  10  books  at  a  certain  price,  and 


16 


ALGEBRA. 


aftcnvards  15  more  at  the  s^xme  rate,  and  at  the  latter  time 

Prlf"-  """  ^'"^  ''  ^^^  ^™-=  -hat.a.th:rS 

Let  a:=the  price  of  a  book  in  shillings ; 
then  10u:=the  price  of  the  1st  lot  in  shiHings 
and    15^=         »  »     2d     "  «     ^  ' 

Now,  if  the  price  of  the  1st  lot  be  taken  from  that  of  th<3 
2d,  there  remains  a  difference  in  price  of  35^ 
.:l5x-l0x=z35s. 
Subtracting  the  10^  from  the  15^,  we  have 

5x=St)s. 

•^='^«.  the  price  of  a  book. 
Frob.  10.  Divide  £300  amongst  A,  B  and  C   so  ihm   A 
Sr^  twice  as  much  as  B,  Snd  C  as'muchVl  tl  B 

Let  ar=B's  share  in  £ 
thch  2.«;=A's  share  in  £ 
and  x+2x  or  3x=zC's  share  in  £. 
I5ut  amongst  them  they  receive  £300  • 
.•.a?-f2.r-f3A—£300, 
0a;=£300  ; 
•  ••  .^=£50  B's  share; 
.-.As  sharc=£100,  and  C's  share=:£]50 

Let  .T= the  number; 
then  9^=nine  times  the  number, 
3.i'=threo  times  the  number, 
and  4.r=four  times  the  number: 
.-.  i^x-}-Sx~4x=48, 
8.i-=48 ; 
.-.   x=z6^  the  number  required 

Let  each  boy's  share =.r ; 
uivu  eacn  woman's  share=r3r 
w>d  each  man's  share  =2  time's  3x=Qx; 


)m  that  of  tli« 


1  as  A  and  B 


SIMPLE   EQUATIONS. 


Hence  we  have, 

the  share  of  the  4  boys     =4. 

<-''^'  «lure  of  the  3  wo,n.n=37inios  'U     o 
,^       and  the  share  of  the  2  inen -o  ,        '  ^'^^■^ 
^iit  the  sum  of  nil  tL       7  ^"^".=^  times  Gj:=12x  : 
sum  of  all  these  shares  is  to  amount  to  £100  • 

..4.r-f9.c+12^=£100,  ' 

25^ =£100': 

.*.  each  boy's  share  — :tU.    nn.h  , 

=/]'>     wl":fl'       t  ^^^oman's=z3  times  £-1 
-^^-  '  ^^^d  each  man's  =  0  tiiues  £4=i:o4 

«hm 'S'  altf t^nr^rtTe''? '  f  r  p™^  ^^--  «- 

third  thrice,  and  to  thV  four  h  f       ?     ^' ^''""  ''^^''^'^'  ^^  the 
'i'-st.     Whai  did  he  give  to  each  J'"'     "''  ''  "^"^''^  ''^^  *«  ^^'^^ 

PKon  14    T.-  ..  ^"''  '^'"  '^^'"  ^^'-^  "^''^  respectively.' 

thatrmfddl^re\,:,l;Tel^bleT  T^^  ^^^^'^  ^-^^-"'^ 
tn-p]^'.  the  least  part.  ^^  *^^  ^''^'^^  '^^"^  the  greatest 

^'^^^^^  5  tinSih:  ^oUr^z  H  ?^  p«'ti  l. 

-ce  between  the  first  ^^^t^::;^!'^^  '"''  li;^  ^>  ^^  f  ^^^"- 

-ixture  contain^  t    ame  quant^'iV  "'  ^  ''^^  "^-     ^i.^ 

"ow  man,,  lbs.  arc  there  of  eS  S  °'  "^^  T*^  T,  *'' 

Pkob.  17.  A  bill  nf  ^nn  -'''«•  8  ibs. 

sovereigns,  and  cmwnsTaS  .Z,  Ijf , '"  ^^^''f'S""'  ''"If 
l-iiid  tlio  numl)cr.  ^"'"  ""'"•'«  of  each  was  nscd. 

Paon  IS   T       .       „  ^■'"''-  400. 

^^.ndfb;d^^;^;™^- ^^  the  same  tin.  n-on, 

Hks  4  miles  an  hour  and  the  n!h     r^  '"^l'"'  ''i'^'^'    the  on. 

^ours  will  they  meet?'  "^^^'^  ^  '"•^«^-     ^i  how  manv 

PiiOT!    10     A  ,  ^'**-  ^  hours. 

luv£lV\u     P^''''"  ^'^"^^^t  a  horse,  chaise    nn  I  f 

I'"   AUO;    the  price  of  the  hor^o  u^no  .    •       ,'  ""'*  harncs-. 

'•■n-ncss,  and  tho'priee  of  the  'Sse  11^''?, ''^'  ^r''  "^'  '^"■ 

h--  and  harness;  what  ^^:^Zl^:^J!:ir'^  ^^  ''<'^'' 


17 


.1 


nswe)\ 


•I'l'ice  of  harness  =  .£'1.3 


<( 


0     8 


2* 


chaise  = 


=   80     0     0 


18 


ALGEBRA. 


•  ■} 

phi! 

i 

I 

iiij! 


SUBTRACTION. 

30.  Subtraction  is  the  finding  the  difference  hctwoen  twc 
filgebraic  quantities,  and  the  connecting  them  by  proper  signs, 
so  as  to  form  one  expression:  thus,  if  it  were  I'equiied  to 
su})tract  5—2  (/.  e.,  3)  from  9,  it  is  evident  that  the  remainder 
would  be  greater  by  2  than  if  5  only  were  subtracted.  For 
the  same  reason,  if  i— c  were  subtracted  from  a,  the  remain- 
der would  bo  greater  by  c,  than  if  b  only  were  subtracted. 
X<nv,  if  b  is  subtracted  from  or,  the  remainder  is  a—b  ;  and 
consequently,  if  b~c  be  subtracted  from  a^  the  remainder 
will  be  a  — j-j-c.  Hence  this  general  Rule  for  the  subtraction 
of  algebraic  quantities,  "  Change  the  signs  of  the  quantities  to 
ha  subtracted^  and  then  place  them  one  after  another,  as  in 
Addition." 

Ex.  1.  From  r)rt-f-3.r— 25  take  2c— 4?/.  The  quantity  to 
•:e  subtracted  ?f<7A  its  signs  changed,  is— 2c-\-4i/ ',  therefore 
the  remainder  is  5a  +  3.c— 26— 2c-j-4//. 

Ex.  2.    From   'rx'—2x  -\-5  take  3.r+5x-l  ; 

The  remainder  is  7.t'— 2.ir  +5     — 3.c^  — S.r+l ; 

ovlx'—Sx'~2x  — 5.r+5   4-l=:4x'^—7x+6. 

But  when  like  quantities  are  to  be  subtracted  from  each 
('ther,  as  in  Ex.  2,  the  better  way  is  to  set  one  row  under  the 
other,  and  apply  the  following  Rule;  ^'■Conceive  the  signs  of 
the  quantities  to  be  subtracted  to  be  changed,  and  then  .proceed 
»i  in  Addition." 


Ex.3. 

From  7a;"— 2.r-f  5 
Subtract  S.i-'^+S.r— 1 


Ex.  4. 

12a'^-3a4-  b-\ 
Oa«+  a-.2b  +  S 


Ex.  5. 


5/y*— 4y-f  3f/ 
0//— 4//—  <' 


Remainder  4.c2—7.r 4-0         6a^— 4a-|-3/>~4     — .?/     *  +4^ 


Ex.  6. 

From  7xr/-\-2x—Sg 
Subtract  S.ri/~  x-^  y 


Ex.  7. 

\4:X-\-y—s- 

x-^y-\-z- 


5 

11 


Ex.  8. 

13.i;8-2.c'^4-'7 


iimkLi^  'iM^i2iLh  a^r^n 


s^i'.  ^V!)at  is  mihtraction  ?    Stntc  the  rule  for  tlio  snbtniotion  of  algcVniic 
mMi.ti'io!^,  and  c.\pl:un  the  pnnoiplc  on  which  it  nsts. 


MVrLE   EQUATIONS. 


19- 


cc  Letwocn  two 
by  proper  signs, 
Bre  I'equiied  to 
t  the  remainder 
ibtracted.  For 
I  a,  the  rernain- 
ere  subtracted, 
sr  is  a— i  ;  and 
the  remainder 
•  the  subtraction 
;he  quantities  to 
'  another,  as  in 

rhe  quantity  to 
h4?/ ;    therefore 


cted  fi'om  each 
}  row  under  the 
ve  the  signs  of 
id  then  .proceed 


Ex.  5. 

6// — 4y —   a 


-r 


-f-4o 


Ex.  8. 

13i;8_Oc'J4-7 


.MutUi 


EM. 


ruction  of  algcVraic  j 


OF  THE  SOLUTION  OF  SIMPLE  EQUATIONS,  C0XTA1N'IX(,  CKLV  OMV 

UNKNOWN  QUANTITY-. 

Rule  II. 

31.  "Any  quantity  may  be  transferred  from  or.e  side  of ' 
;.e  equauon  to  the  other    oy  changing  its  sign;"  and  it  il 
toundea  upon  the   axiom,   that  "if  equals   b?  added  to  ^r - 
equal"  '^"'^''   '^''    '""''   ^^   remainders  Zm   be  • 

Ex.  1.  Let  ;r+8^15;  subtraci  8  from  each   side  of  the 
equation,  and  it   becomes  :.+8-8  =  15-8j    but  8-8^0 

0,,f;iV^^'l-^=3»;  f<'7  to  eachside  of  the  equation,,, 
mciii     7+7_,i0+, ;  but  -7+7=0;  .-.^=20+7=27. 

tnenrf.r-2.r=o^_o^.4.9^_5     but  2.r-2j;  =  0    •  3ar-'>r-Qv^ 
+  5.     Now  3^-2^=;.,  and  9+5  =  14;  hence  ;:!  14. 
On  reviewing  the  steps  of  these  examples,  it  appears 
(1.)  That  a; 4-8=15  is 'identical  with  a;=15_8 

(2.)     «     ;r-7=20  «        with  a:=204-7* 

(3.)     «    3.r-5=2x+9       "        with  3.r-2.r=9-f  5. 
Or,  that  "the  equality  of  the  quantities  on  each  side  of  thp  - 
equation,  is  not  affected  by  removing  a  quantity  from  one  s      ^ 
of  the  equation  to  the  o-:her  and  chauffinr^  its  s^nP 

he  Idl  out  ff \  1-  ""''  ^'f'  '''^'''  ^^  ^»  ^q^'-ition,  it  n)ay  . 

I'e.  lUt  out  of  the  equation;  thus,  ii'x+a=c+n  then  x-cl 
'^~a;  huta~a  =  0,,\x=c.  ^tu,  men  x^c-^- 

h  further  appears,  that  the  simis  of  all  th.  *««.„.  ^c> 

>n  may  be  changed  from  -j-  to -or  from  "  "in  T  ""."^'i"'*; 

«,  .hen,  by  the  l.ule,  .r^c-a+i;  change  the  siirns  of 


■) 


lion 


e  sijijns  of  alt 


•  mm 


20 


ALGEBRA. 


Ex.5. 
Ex.  0. 
Ex.  7. 
Ex.  8. 
Ex.  9. 
Ex.  10. 


-a  \-C  =  Z,  or  T7: 

Ans.  xz=l4. 
Ans.  rrz=29. 
Ans.  x~G. 
Ans.  a;=9i~-2a. 
Ans.  x=2. 
Ans.  ar=10. 
A71S.  ar=5. 
-.4;t5.  a: =3. 

a 


'  Ihe  terms,  then  b   -x-,-,,  -r,  in  wliich  ca^c  b 
t—a-\-b,  as  before. 

Ex.4.       2i;+3=a:4-17. 

5.K— 4=4ar+25. 

7ar-9=e».r— 3. 

4x+2a=3a;+9i. 
15^+4=34. 

8a;+7=6.r4-27. 

9,r-3=4.r4-22. 
Ex.  11.  17x-4jr+0=3i,'+39. 

Ex.  12.     ax—c=b-{-2c. 
Ex.  13.     5^-(4.r-G)  =  12. 

The  sign  »—  before  a  bracket  being  the  sign  of  the  wholf 
quantity  enclosed,   indicates  that  the  quantity  is  to  be  sub. 
traded;  and  therefore,  according  to  the  Rule,  Avhen  the  brack- 
ets are  removed  the  sign  of  each  term  must  be  chantrcd 
rLiis,  the  signs  of  4.r  and  of  C  are  respectively  4-  and  — ,''but 
when  the  brackets  are  removed  they  must  be  chann-ed  to  — 
:tind  -f  respectively.     The  equation  then  becomes 
5^-4.r+G  =  12. 
By  transposition,  5x—4x= 12— 6; 

.-.  .r=6. 

Ex.  14.  6.r--(8+.r)=4.r-(x-10). 

By  remo^'ing  the  brackets,  and  changing  the  signs  cf  the 
.  terms  which  they  enclose,  the  equation  becomes 

Ox— 8— a:=4=:4,r— .r  +  10. 
Transposing,  Gx~z—4x-\-z=l0-^8 ; 

.-.  2x=18.     .. 
Dividing  botli  sides  of  the  equation  by  2, 

Ex.  15.     4x—{Sx-\-<)=S. 

Ex.  la     Si;-(Cu'-8)=9-(3-T). 

Ex   17.     4x~(Sx-Cy)~(4r-m  =  l9 

Ex.  18.     5a:--(8-l-3r)  =  8^(-ir— 1). 


Ans.  .r=::12. 
Ans.  x=-^2. 


-  /-.,_ 

If.. 

Ans. 

x~ 

:2. 

Ans. 

x  = 

12 

SIMPLE   EQUATICXS. 


21 


!  sijnis  f  f  the 


^9. 

X  — 

12. 

IS. 

x=. 

-% 

IS. 

!  (it 

}■ 

2. 

IS. 

X  — 

12 

PR0JJLKM8.  ■ 

Let  therefore  x=.m^  Jess  number; 

then  will  a:+15  =  the  greater: 
iJut  their  sum =59; 

.-.  a;+a;-rl5— 59,' 
or2.r+15=,59.  ^ 

And    transposing  15,  2.r=59-15, 

or2.r=44; 

.-.  ar=:22  the  less  number 
-rxo — ^^-i-io~j7  the  greater. 

J3ut  together  they  receive  27  ;  '^''^  * 

.*.  .r+a:+5=:27, 
.,,  .  or2.c+5=27. 

lrans]-/,Kwig,  2.r=:27-5, 

or2.r=:22; 

••.  .r=ll,  the  No.  James  receiver] 
.6  -r  o  _  1  o  Kichard  received 

Let  a-rrthe  number; 
then  4.r=4  times  the  number, 
.  2.c= double  the  number, 

.h.refoie,  b;y  the  equality  stated  in  the  question 

_        4a;=2ar+12.  ^ 

»,y  trjpsposition,  4.ir— 2;r=:]2. 

Oj. TO  . 

.'.  arr=  6. 


•aon.  4.  At  a,  election  420  persons  voted,  and  th 


le  succcs.* 


i 


ill 


*|!1 


1' : 

! 
iilli 


22 


ALGEBRA. 


ful  candidate  had  a  majority  of  4(5. 
each  candidate  ? 


How   many  v<itcd  for 
Ans.  187  and  233. 

Prob.  5,  One  of  two  rods  id  8  feet  longer  than  the  other 
but  the  longer  rod  is  three  times  the  length  of  the  sh(.)rter. 
\\  hat  arc  their  lengths  ?  Am.  4  H.  and  12  ft. 

Prob.  0.  Five  times  a  number  diminished  by  16  is  equal 
to  three  times  the  number.    AVhat  is  the  number  1   Ans.  8. 

^  Prob.  7.  A  horse,  a  cow,  and  a  sheep,  were  bought  fur 
£24 ;  the  cow  cost  £4  more  than  the  sheep,  and  the  horse  £10 
more  than  the  cow.     What  was  the  price  of  the  sheep  ? 
Let  a:=:thc  price  of  the  sheep  in  ^; 
then  .r4-4=:  "         "     cow         •' 

and  a;+4+10=  "         "      horse      " 

But  these  thi-ee  prices  taken  together  amount  to  £24 ; 

^    .-.  a:-f(ar+4)4-(a:-f-4+10)=24. 
Adding  together  lilic  terms, 

3.^4-18=24. 
By  transposition,  3a? = 24  —  1 8, 
Sx=Q  ; 

.-.  «=£2,  the  price  of  the  sheep. 
Prob.  8.  A  draper  has  three  pieces  of  cloth,  which  togethci 
measured  159  yards;  the  second  piece  was  15  yards  iongei 
than  the  first,  and  the  third  24  yards  longer  than  the  second. 
What  is  the  length  of  each  piece  ] 

Ans.  35,  50,  and  74  yds. 
Prob.  9.  Divide  £3G  among  three  persons.  A,  B,  and  C,  in 
such  a  manner  that  B  shall  have  £4  more  than  A,  and  0  £7 
more  than  B.  Arts.  £7,  £11,  and  £18. 

Prob.  10.  A  gentleman  buys  4  horses  ;  for  the  second  he 
gives  £12  more  than  for  the  first;  for  the  third  £5  more  thun 
for  the  second ;  and  for  the  fourth  £2  more  than  for  the  third. 
The  sum  paid  for  all  the  horses  was  £240.  Find  the  price  oi 
'^Jach.  ^^s.  £48,  £G0,  £65,  and£67. 

Prob.  11.  What  nimibcr  is  that  whose  double  is  as  much 
above  21  as  it  is  itself  less  than  21  ? 

Let  a;=:the  number ; 
then  2.r= double  the  iiiunber, 
2.r— 2i=what  double  the  number  is  abo/e  21, 
and  11  — a:=what  the  number  is  lesi  than  21 : 


nany  vcted  fo? 
187  and  233. 

than  the  other 
of  the  shorter. 
11.  and  12  ft. 

by  16  Is  equal 
)er'?    Ans.8. 

ere  bought  for 
I  the  horse  £10 
le  sheep  ? 


t  to  £24 ; 


ecp. 

which  togethci 
5  yards  Jongei 
lan  the  second. 

and  74  yds. 

A,  B,  and  C,  in 
I  A,  and  0  £7 
11,  and  £18. 

the  second  he 
I  £5  more  tliun 
in  for  the  third, 
ind  the  price  oJ 
55,  and  £67. 

j\e  is  as  much 


!  al)o/e  21, 
an  21  ; 


SIMPLE   EQUATIONS.  .  23 

By  transposition,  2j:+a:=21  +  21     ^' 

3u:=42'; 
rpi  .'.  .r=:14. 

ine  answer  may  easilv  be  nrnvn.7  f^  r 
-28-21  =.7,  and  21^-2^^141^7     T'-^'^'"'^ 
a^  n.uch  above  21,  as  21  is  above  HJainei;^;''  ''''''  ^'^  '' 

»5  to  spare,  and  bv  ^wZ  \fl  n%  ^  '"t  f  ^''  ^'^^^  ^^V  I  Had 
How  Lny  boysV?;;  I'e?  "''  ""''  '  ''''''  ^'^  ^---^g- 

^hpnJfT       ^^*/=the  number  of  bovs; 

^hen.  If  I  gave  to  each  boy  4  oranges"'  I  \hr..,u^     • 

times  X  oranfres-  "^^I'lfets,  i  should  give  away  4 

.;.  ro/«/  «nwifr  of  orangcs=4^4-fi  . 
Agani,  if  each   boy   receivorl   r^-*'*'^^- 
o-angcs  left ;  ^    rtceucd   3   oranges,  there   were   12 

terms  of.  must  necessarily  C::^:afi::Zll^''^''''''  '" 
o    ,  .     •'•4:i^  +  C=3,r+12.  ^  '^ 

I5y  transposition,  4x-~Sx=l2~0- 

.:  x~G^  the  number  of  boys. 

I>ut  hrcmlstitLTK  T  ?"'  '"  ''''''^  "40  miles  in  4  dav. 
He  murg?rmnes  he  sl"Tf '^'?''^''"  ^^""^^  ^^at' 
fourth  da.^  less  t  ai  the  S"  it^'  '  '"  '^^'■''  ''^"^^  ^^  ^^e 
travel  each  day?  '     ^^""^  "^^"3^  ^"'^es  must  b- 

then^rllZ^'^^  number  of  miles  on  the  1st  day 

,    ^-9=  «  «  «      ?^ 

and  .r- 14=:  u  „  ^^      ^^ 

4th 


.^T  the   number  of  miles  whioh   hn  I'l 


a 


13     lU 


aajg 


^+x~.5+x~9+x~U~2. 


Collecting  the  terms,  4x  -28- 2 


40. 


240. 


u 


ALGEBRA. 


Jjy  transposition,         4a;=240-f28, 

4.r=2G8; 
.'.  ar=07,  the  number  of  miles  he  goes  on  1st  Oav, 
ar-5  =  C2    "  "  "         "         "      2d     " 

g. 9—58    "         "  «         "         «       3(]     " 

and  a; -14=53    "         "  «        "        *      4th  '• 

Prob.  14.  It  is  required  to  divide  the  number  99  into  five 
such  parts  that  the  first  may  exceed  the  second  by  3,  be  less 
than  the  third  by  10,  greater  than  the  fourth  by  9.  and  Itss 
than  the  fifth  by  10. 

Ans.  The  parts  are  17,  14,  27,  8,  S3 

Prob.  15.  Two  merchants  entered  into  a  speculation,  by 
which  A  gained  £54  more  than  B.     Tlie  whole  gain  was  £49 
less  than  three  times  the  gain  of  B.     What  were  the  gains  ? 
J  ^;is.  A'sgain=£157;  B's=£103. 

Prob.  10.  In  dividing  a  lot  of  apples  among  a  certain  num. 
ber  of  boys,  I  found  that  by  giving  0  to  each  I  should  have  too 
few  by  8,  but  by  giving  4  to  each  boy  I  should  have  1 2  re- 
maining.     How  many  boys  were  there  1  Ans.  lOi 


MULTIPLICATION. 

32.  Multiplication  is  the  finding  the  product  of  two  ot 
more  algebraic  quantities ;  and  in  performing  the  process,  the 
four  following  rules  must  be  observed. 

(I.)  When  quantities  having  like  signs  are  multiplied  to. 
gether,  the  sign  of  the  product  will  be  +  ;  and  if  their  signs 
are  unlike,  the  sign  of  the  product  Will  be  —  .* 

*  Tlii.s  rule  for  tlie  multiplication  of  the  Signs  may  be  thus  ex- 
plained : — 

I.  If  -fa  is  to  be  multiplied  by  +ft,  it  means,  that  -\-a  is  to  be  added 
to  itself  as  often  as  there  are  units  in  6,  and  consequently  the  product 
will  be  -f  ab. 

IT.  ]f  — a  is  to  be  multiplied  by  -^b,  it  means,  that  — a  ih  to  ho 
added  to  itself  as  often  as  there  arc  units  in  b,  and  therefore  the  product 
}b  — ab. 


f5<>.  "What  is  multiplication,  and  what  are  the  Rules  to  be  observed  In 
multipl  cation  i 


I 


MULTIPLICATIOX. 


25 


Mt 


(2.)  Tlic  coefTidcnts  of  the  facfor^  mu^t  Im  r,,.,ii-  i-    . 
g«hcr,  ,o.brm  the  coefficient  o?Zp'o2cl  "'"^'^""'  '" 


forming  the  operation,  the  kule  i,  "  Tn  n  ,u-  ?     J    P° 
..gns, ./,«  the  coefficic'its,  J^^^ar^: ^I^H^J-''  '^ 


r- 

the 


ay  be  thus  ex- 


be  observed  In 


Case  I. 

^^'■il'lTJtX^,:::.  «'"^"  ^™--"-  ^  '■o-  wi.ich  the 

U  -  »J.  ""'  """*  "■  '.  ""tl  ~n»equently  tl,o  proiluct 

Or,  those  four  R.le,  ,„i„.,,t  b„  ,„  e„™pr„,,„„jej  ,■„  „„,  .  „,^, 

I  J?.s"i;r'/i- '„t;w;L'3r;'  :^4'ef."-  »^"  -  *- 

But  a  — .  /j,  arfc?<'(/  c  ^•»^^ft«-  .  ,      =  a^  —  ^r 
and  a  —  6,  subtracted  d  times    =  __  «^  |:  f,j 

i.  e.,  -f-  a  X  +  c  =--  4-  «/• 


+  «  X  — (/  =  — 


ad 


—  bX~d  =  i.bd. 


Ex.  1. 
i)a 


Ex.  5. 

Aabc 
3ac 


ALG 

EBRA. 

Ex.  2. 

Ex.3. 

— Oax//* 

-15aV/'c 

Ex.  6. 

Ex.  7. 

dxhf 

— 4c*f^ 

—2?/ 


2c 


Ex  4. 

—  5a*4c 
-~_2iV 

4-lOaV/cl' 


>:x.  8. 

— 7c/.r*?/ 
— 2f/A 


/J^-^/b 


^^^ 


^cW^ 


Case  II. 


34.  When  one  factor  is  compound  and  the  other  simple^ 
''Tlien  mc/i^erm  of  the  oompound  factor  must  be  multiplied 
by  the  simple  factor  as  in  the  last  Case,  and  the  result  wil' 
be  the  product  required." 


Ex.  1. 

Multiply  Sab—2ac-^d 
by  4rt 


Product  V2a^b—Sa^c+4a(f 


Ex.  2. 

Sx^  — 2x'-!-4 

—  14aa? 

—  42ax~-\-  28a^?^5«^- 


Ex.  3. 

Multiply         Ix^  — 2.C  +4a 
by   —  3a 

FVoduct 


Ex.  4. 

12a^-2a'  +  4a-l 
Sx 


-21a.r'-f-(k,';c— 1-.V 


r 


Ex,  5. 

Multiply      dah-^3a—x-^] 
by  —  x^ 

Product   ZJ^,^^  ^  y^^t- 


[  Ex.  6. 

!        4a;V  +  3.r— 2i> 
— 3.r?/ 


Case  HI. 

35.  When  both  factors  are  compound  quantities,  each  term 
of  the  multiplicand  must  be  mu'tiplied  by  each  t  erm  of  the 


£x  4. 

—  ha*bc 


>:x.  8. 


MULTIPLICATION. 


27 


multiplier;   mid  then  placing;  Ike  qnantitu-  nmhr  ea,\  ot\t» 
Uie  sum  of  all  the  terms  ^viJl  be  the  produo.  required. 


Ex.  1. 

Multij  ]y  a  -I-  h 
by  a  -f  h 
1st,  by  a  qS^-  ah 
2d,  by  i  /7A-i-// 

Product  aH^oHhO"" 


Ex.2. 


a" 


*   -/^« 


Ex.  a 


fl. 


3      i/c 


D  Other  simple, 
:  be  multiplied 
the  result  wil' 


J  A*    /s^* 


-2a;»-h4 


+28a.r^-50c/- 


Ex.  4. 


Ex.  G. 

4a;V+3.r— 2^ 


ities,  each  term 
ach  '  erm  of  the 


Ex.  4. 

3.r*+  2ar 

4.r  +  7_ 

i2?+~a? 

±?lfM-14T 

12.cH29x^-fT4j 


Ex.5. 
3.C'-  2x  +5 


•-21.g^-f-14.r-35 
18.r''~33.r2+44,r"-^^ 


Ex.  C. 

14a  c  —  3a  *  -f  2 
a_c  —     ah  -}-   I 

-Ua'bc            +3a%'^2ab 
•hiiac     —  Sab + 2 

— !tiZ!f!^±l?^+3a-^A«'^(^:f2" 


Iv  -i-2 


Ex.  7. 


jr 


+  2^.*^--  ar+4 


+ 


IT 
~4 


Ex.  8.  Multiply  a'^+3a«34-3a/y^+63      ^y      a-^b. 


28 


AliGEBllA. 


Ex.  9.   Multiply  4x'?/-f-3xy— 1   -  -  Ly  2.c'— a-. 

Ex.  10.       «        a^—x^^x-b  -  -  -  by  2x'-\-x-\-l. 

Ans.  2x'—x*-^ 2x''-l0x^—4x^b, 

Ex.  11.       «         3a«-f-2a6-i2 by  Sa'^-2ab-i-lj\ 

Ans.  9a'-4a^lj*i-4aP-bK 

Ex.12.       «         ar'+xV+^y'+y'     by  a;-?/. 

-.4ns.  x*—y*. 


Ex.13.       "         ar'-Jar+l 


by  x'^—lx. 


A /IS.  a;'— |-.c^+y.r— ^.r. 


ON   THE   SOLUTION   OF  SIMPLE  EQUATIONS  CONTAINING  ONLY  0NJ5 
<  UNKNOWN   QUANTITY. 

Ex.1.     3.c-}-4(jr4-2)=3G. 

The  term  4(a;+2)  means,  l  ,.it  x-\-2  is  to  be  multiplied  by 
4,  mid  the  product  by  Case  2d  is  4j;+8; 

.-.  3x-^4x-\-8=S6. 
Adding  together  the  terms  containing  x,  and  transposing  8, 

•7x^m-S, 
7a;r=28; 

•    •   •C'  ■^—  Htm 

Ex.2.     8(a;+5)+4(T-f  1)=80. 
Perfurming  the  multiplication, 

8^+404-4.c+4=:80. 
Collecting  the  terms,  12x4- 44 =80. 
Transposing,  12.f=80~44, 

12x=:30; 

Ex.3.     C(.r4-3)+4.ir=58.  Ans.  xt^4. 

Ex.  4.  30  (x_3)4-6=:0  (.t-f  2).  Ans.  x^i. 

Ex.5.     5(.i;4-4)-3(«-5)=49.  Ans.  x=:l 

Ex.6.     4(3+2.r)-2(0-2.i-)  =  60.  Ans.  x-^5 

Ex.7.     3(x-2)4-4=-.4(3-.r).  ^«*.  .r^^o; 

Ex.  a     0{4-a-)  -4(G-2,r)-12=0.  Ans.  a-^O. 


SIMPLE   EQUATIONS. 


29 


riNO  ONLY  ONfi 


PR0DLEM8. 

suin  shall  be  35  ?  '''''^  ^^  ^  ^""^«  the  less..  tJie 

Let  ;r=:tho  less  numLer  • 
\n^  q/-        ,,^^^"-^+9=thc  greater.         ' 

^^.t  by  the  problem,  3  times  the  greater  +  5  times  the  le.. 

.-.  3.r4-274-5.r=35, 
.  8.r+27.-=35. 

Iransposmg,  8.r=35-27=8; 

.-.  ,-r=:  1 ,  the  less  number, 
and  .r+9=10,  the  greater. 

.'•"d  in  order  to  do  thif  o'"i.  ol         w''"*  ^^  ^^^'^^'^*^  ^^i"^- 
^-r:    |uho.man;t,i:j::^K,:-^ 

then^tS""  """'"  ^'^?""  ^^  ^^^-^> 
land  7(.;+5)'^'!^'  """"^^'^  "^"^''^^  the  "M       " 

|num^.'^i,:'fP^^'^'^'"  ^'^  ---,;;  b!^LrLl  the  same 

.'.  bi-=7(^+5), 
^         12.c=:7,c+3o. 
iransposmg,     12.r-7.r=r35, 

^on-ier  is  in  overtaking  tL'f^'sf  ""n.b.r  of  hours  the  .ocu,rl 

kJlr^t^etLtS^;;^^^ 

Mow-  nnmy  passengers  were  ty/c  oTleh  dllf  "^'"''   ^^^ 
Let  «= the  numl)er  of  nas«nnr,<>.., 

6a' = 


4>1C*      ;-vi*    f  1-.-        "i   ,,i.       _» 


kn.J 


.  p  .      ,  "      2d       " 


Mi 


3* 


tiutmitmi 


SiSkd 


pl! 

1' 

hi 

1 

1 

(■■ 

1 

5<' 


ALGEBRA. 


But  these  two'sums  amount  to  £3  12s.,  or  to  72«. 
.-.  6.r+4(15-rr)-72, 
6x-\  (30— 4.r=72. 
Dy  transposition,  Gu;-- 4a; =72— GO, 

2.r  =  12; 

.'.  x  =  G  No.  of  1st  class  passengers  ; 
/.  the  number  of  2d  class  pas8engers=15— a;=9. 

Prod.  4.  What  number  is  that  to  which  if  G  be  added  tvvic€ 
the  sum  wi'l  be  24?  Ans.  C. 

Piion.  5.  What  two  numbers  are  those  whose  difference  is 
0.  and  if  12  be  added  to  4  times  their  sum,  the  whole  will  be 
GO?  '  Jns.  3  and  9. 


Prod.  0.  Tea  at  6s.  per  lb.  is  mixed  with  tea  at  4.9.  per  lb., 
and  161bs.jof  the  mixture  is  sold  for  £3  18s.  How  many 
lbs.  were  there  of  each  sort  1  -.-Ins.  7  lbs.  and  9  lbs. 


Prod.  7.  The  speed  of  a  railway  train  is  24  miles  an  hour, 
and  3  hours  after  its  departure  an  express  train  is  started  tc 
iHUi  32  miles  an  hour.  In  how  many  hours  does  the  cxpres;-' 
overtake  the  train  first  started  1  Ans.  9  hours. 

Prod.  8.  A  mercer  having  cut  19  yards  from  each  of  three 
equal  pieces  of  silk,  and  17  from  another  of  the  same  length, 
0)nnd  that  the  remnants  taken  together  measured'  142  yard^. 
W  hat  was  the  length  of  each  piece  ? 

Let  a:=the  length  of  each  piece  in  yards  ; 
.*.  X — 19  =  the  length  of  each  of  the  3  remnar.ts, 
and  .r— 17= the  length  of  the  other  remnant; 
then  3  {x-l9)-^x-]1=U2, 
o^3a;-57^•a:-17=:142, 
4.r-74  =  142. 
Transposing,  4.r = 1 42 + 74, 

4.c=216; 
.',  .'rr=54. 

Prod.  9.  Divide  the  number  G8  into  two  such  parts,  thntl 
the  ditference  between  the  greater  and  84  may  equal  3  timesj 
the  diubretit*  Dotween  tno  less  an.u  40. 
Let  ir=the  less  part, 
then  G8— ar=thc  grsfiter; 


SIMPLE  EQUATIO:  ^, 


^ .  |4oi:lLi't:ri^!,v^!i«^  ?■■  ^  «.o  ...at.. 


31 


n.n  b,  '-."X^^e^^f^  -  e,ua.  .  each  .W . 

By  transposition,    ^+3^=120+68-84 

4a;=I04;  ' 

and.-.thegroater=42:""''''P"''' 

to  two'upon  etl^Tal  "CL"'  <=^.■•'^^ Netted  three  shillm«, 
•tags.    How  ma„;  deais  mV^^^f  "'"'^  "«  ^™»  Ave  sli 

..•:0-:r=the  number  he  lost; 

iosfwafl,""^^"-^^  ''^'-»  *e  -no,  won  and  the  money 
.•.2^-3.  (20-.r)=  5 
2^--60+3ar=  5,' 
5x-.Q0=  5, 

.'.  «=13. 

to  i^iff  Lt 'tfan^t^^^^^^^^^      P^^y^^l  packs  of  cards  so  as 
off  twice  as  many  as  B  ill  and  R  .T  ^  happened  that  A  cut 

'^'"^|;^=the  number  he'lefl, 
.  4 ^^=^,*Jo  number  B  left; 

3"t  the  number  Bci^n^  """'^^^  ^«  «"*  <^ff^    ' 
^  ,,ft .  "^^^r  B  cut  off  was  equal  to  7  times  the  number 

.••52-a.=7.(52-2z) 
52— ar=364--14ar 
transposing,  |4;r-ar=364~-5o 
la^=312;       ' 
a:=24 


i 


Aou  off48,  and  B  cut  off  28  cards. 


A^-ms 


m 


S2 


ALGEBRA. 


pROB.  12.  Some  persons  agreed  to  give  sixpence  each  to  a 
waterman  for  carrying  them  from  London  to  Greenwich;  but 
with  this  condition,  that  for  every  other  person  taken  m  by 
the  way,  threepence  should  be  abated  in  their  joint  fare. 
Now  the  waterman  took  in  three  more  than  a  fourth  part  of 
the  number  of  the  first  passengers,  in  consideration  c.4*  whicli 
lie  took  of  them  but  fivepence  each.  How  many  persons  were 
there  at  first  1 

Let  4x  represent  the  number  of  passengers  at  first ; 
then  3  more  than  a  fourth  part  of  this  number = a: -f  3,  and 
they  paid  3  (a? +3)  pence. 

.-.  the  original  passengers  paid  6x4a;— 3(a;+3)  pence. 
But  the  original  passengers  paid  5x4a:  pence ; 
\  by  equalizing  these  two  values,  we  get 
6x4a?-3  (a:4-3)=5x4ar, 
i      24a; -3a; -9= 20a;. 
Transposing,      24a; — 3a; — 20a; = 9 ; 

.  •  Xzz^tj  ; 

and  .'.  the  No.  of  passengers  were =4x9 =30. 

Prob.  13.  There  are  two  numbers  whose  difference  is  14, 
and  if  9  times  the  less  be  subtracted  from  6'  times  the  greater, 
the  remainder  will  be  33.     What  are  the  numbers '? 

Ans.  17  and  31. 

Prob.  14.  Two  persons,  A  and  B,  lay  out  equal  sums  of 
money  in  trade;  A  gains  £120,  and  B  loses  £80;  and  now 
(\.'s  mo^iev  is  treble  of  B's.     What  sum  had  each  at  first  1 
^  Ans.  £180. 

Prob.  15.  A  rectangle  is  8  feet  long,  and  if  it  were  two  feet 
broader,  its  area  would  be  48  feet.     Find  its  breadth. 

Ans.  4  feet. 

Prob.  16.  William  has  4  times  as  many  marbles  as  ITiomas, 
but  if  12  be  given  to  each,  William  will  then  have  only  twice 
lis  many  as  Thomas.     How  many  has  each  % 

Ans.  24  and  6. 

Prob.  17.  Two  rectangular  slates  are  each  8  inches  wide, 
but  the  length  of  one  is  4  inches  greater  than  that  of  the  othw. 
Find  their  lengths,  the  longer  slate  being  twice  the  area  of  the 


J  each  to  a 
nivich ;  but 
taken  iii  by 
joint  fare, 
th  part  oi' 
)n  (.f  whicli 
jrsons  were 

first ; 
=x-\-S,  and 

3)  pence. 


rence  is  14, 
the  greater, 

rand  31. 

al  sums  of 
) ;  and  now 
at  first  1 
ns.  £180. 

ere  two  feet 

adth. 

IS.  4  feet. 

3  as  ITionias, 
e  only  twice 

24  and  6. 
inches  wide, 
of  the  othcj'. 
e  area  of  the 


SIMPLE  EQUATIONS. 


83 


hZZf"'  ^''''  ""^  ^  '''*^^^'  ''  '^  ^'"S^^  multiplied  by  its 

Rnf  :i:  ^f  ^"^  ^  ^'^'t'*)  ''''^  ^^^  ^^^as  of  the  slates. 
But  the  larger  slate  is  twice  the  area  of  the  less 
.•.8a;X2=8(a;+4), 
16x=8x-{-S2; 
.'.  8ar=32 ; 

.••ar=  4,  the  length  of  the  less  slate, 
and  0^+4=  8,    «        "         «     greater  slate. 

h  71'  ^f'.x^"'"'  rectangular  boards  are  equal  in  area  •  the 
breadth  of  the  one  is  18  inches,  and  that  of  the  other  16 
mches,  and  the  difference  of  their  lengths  4  inchef  F^d  tJe 
length  of  each  and  the  common  area. 

p         ,^    ,         .  ,  ^^*-  32, 36,  and  576. 

Frob   19.  A  straight  lever  (without  weight)  support^  in 
equilibrium  on  a  fulcrum  24  lbs.  at  the  end  of'the  shorter  arm 
and  8  bs.  at  the  end  of  the  longer,  but  the  length  ofthTS 

r     .1.    %r^^'  ""^'^  *^^  ^^^t  «^  the  shorter.     Find  thi 
lengths  of  the  arms.  "  ^"® 

Let  ^=length  in  inches  of  the  shorter  arm  • 
thenar+6=     -  .«  »       j  »  ' 

Now  the  lever  will  be  in  equilibrium,  when  the  weiVht  at 
one  end  multipl  ed  by  the  length  of  the  correspond i^Sm  ?s 

Xg  InnT"'     ''        '''"  "''  "^^'^^^^^  b^  it--reL 
.-.  24a; =8  (x-\-e>), 
24a; =8^+48, 
16a;=48; 

V  f =3  -inches,  the  length  of  the  shorter  arm  • 
anda;+6=9      "        "      »        «       longer    '^  * 

Pkob  20.  A  weight  of6  lbs.  balances  a  weight  of  24  lbs  on 
H  lever  (supposed  to  be  without  weight),  whose  lenSh  is  20 
.r^hes ;  if  3  lbs.  be  added  to  each  weight;,  what  addittn  must 
be  made  o  each  arm  of  the  lever,  so%hLt  the  fulcrum  mav 
Ed?  '^'''"^  ^'"'""'  ^"^  equilibrium  Btm  be  ;i: 

This  problem  resolves  itself  into  two  other  problems  :~. 

:■..■    --.-.uru    uic;  i.:m,r,r,.=.  ..r   rr..   „„    j^  ^-^^  Original   posi. 


<ion 


iCiii-tna  OI  iiitj  an 


Let  a;=the  length  in  inches  of  the  shorter 

JO— ar=   "         "  "  «      1 

lonffer 


long( 


arm: 


X. 


u 


r% 


I  Mi 


!i 


\l 


B4: 


ALGEBRA. 


Now,  in  order  that  there  may  be  equilibrium.  24ar  and  6  (20 
—a;)  must  be  equal  to  each  other ; 
.•.24a; =120— 6a:, 
30a;=120; 

.*.  x=     4,  the  length  of  the  shortei  arm; 
and20— ar=  16,    "        "        "      longer    " 

(2.)  To  find  the  addition  to  be  made  to  each  arm,  so  that 
there  may  again  be  equilibrium  on  the  fulcrum  in  its  original 
position,  after  3  lbs.  have  been  added  to  each  weight : 

Let  X  —  number  of  inches  to  be  added  to  each  ai  m  ; 
then  the  lengths  of  the  arms  become  4+ a;  and  16 -fa;  inches 
respectively :  and  the  weights  at  the  arms  have  been  respect- 
ively  incre"=ed  to  27  lbs,  and  9  lbs. 

But  by  i..e  principle  of  the  equilibrium  of  the  lever,  27  (4 
-f-a;)  and  9  (16+a:.),must  be  equal  to  each- other; 
.•.27(4+a;)=9(16-fa;). 
Divide  each  side  of  the  equation  by  9,  and 
3(4-|-a:)  =  16+a;, 
12+3^"= 16 +a;, 
3a;-a;  =  16-12 
2a;=  4;        * 

Prob.  21.  The  condition  being  the  same  as  ih  the  lass 
problem,  how  many  inches  must  be  added  to  the  shorter  arm 
in  order  that  the  lever  may  in  its  original  position  retain  its 
equilibrium'?  Ans.  1^  inch. 

Prob.  22.  A  garrison  of  1000  men  were  victualled  fdr  30 
days  ;  after  10  days  it  was  reinforced,  and  then  the  provisions 
were  exhausted  in  5  days ;  find  the  number  of  men  in  the  re- 
inforcement. Ans.  3000. 

Prob.  23.  Two  triangles  have  each  a  base  of  20  feet,  but 
the  altitude  of  one  of  them  is  6  feet  less  than  that  of  the  other, 
and  the  area  of  the  greater  triangle  is  twice  that  of  the  less. 
Find  their  altitudes.  Ans,  6  and  12. 

N.  B.  The  area  of  j^  triangle  =  ^  base  X  altitude. 


D     -  - 


A 


_  _      1      Ti 

iinU    JD 


uc^aii.  i,o  I'iiij    yviLii  equaj  auras ;    n 


won  VZs. ;  then  6  times  A's  money  was  equal  to  9  times  B's, 
What  had  each  at  first?  Ans.  £S. 


X  and  6  (20 


31  arm; 


•m,  so  that 
ts  original 
It: 

3ach  ai  m ; 
i-x  in(;hea 
m  respect- 
aver,  27  (4 


I  the  lasr, 
lorter  arm 
:  retain  its 
[^  inch. 

ed  for  30 
provisions 
in  the  re- 
.  3000. 

)  feet,  but 
the  other, 
f  the  less, 
and  13. 
de. 

aUiria  j    A 

times  B's. 
ns,  £S^ 


SIMPLE   EQUATIONS.  g^ 

rnvT^hut^t  ''^f  ^  'l"^^"g  ^^^^^  reckoning  at  a  tavern 

' .,      ,     wcUKs  Ai  miles  an  hour  morp  than  \      rr^ 
many  m.le.  does  A  walk  i„  an  h  .r?"  'its  milS 

DIVISION.  -^ 

:atl„%*f  ''""'=  «>»•  f°"»-tom.draWyfr„„  tl.at  in  Mul.ipli 
If +«X+4=+„S.  then +1*=+J,  „„d  ±±=+^ 


4  a  X-b^-ab,  -  -  -  =1^  =_6.  and  =11*  ^^a 


— aX~6=+a5,  - 


—  a 


»■  *.,  /j/lrff  eigna 
•  produce-f ,  and 
unlike  signs — ^. 


■f  -L 


iltj 


ButtaXLT""  '''""'  "'"'"""  o'lS"""^"  9»"ti,i«,(    s,...  th. 


SQ 


ALGEBRA. 


In  the  divisor  must  be  subtracted  from  its  index  in  the  dividend 
to  obtam  its  index  in  the  quotient.     Thus, 

(1.)   +abc  divided  by  +ac  -  .  or     ±^     =4-5 

4-ac 

(2.)  +Qiibc   -  .  -  -  .  -2a  -  -  or 

(3.)  —lOxi/z    .  +5y    -  or 


6abc 


~2a 
— lOxyz 
+5y 


=  —36c. 
=z—2xz. 


(4.)   ~20a'A3 4a^y  or  =|?.^' =+5aa:/. 

Of  Division,  also,  there  are  three  Cases :  the  same  as  ia 
Multiplication, 

'         Case  1. 
37.  When  dividend  and  divisor  are  both  simple  terms. 


Ex.  1., 

Divide  18aa;«  by  Sax, 


IQax'- 


=6a?. 


Ex.2. 

Divide  15a'b^  by  -5a. 

+  15a''i« 


3aa; 

Ex.3. 
Divide  —28a^y^  by  —  4zy. 

Ex.5. 

Divide  —  14a^6'c  by  7ac 
-14a«6'c 


— 5a 


=  -3a6«. 


-h7ac 


Ex.4. 

Divide  25aV  by —5a»c. 

+25aV 
— 5aV      "~ 

Ex.  G. 
Divide  —  20a;y2'  by  — 4y«. 
~20a;'yV 


— 4y2. 


Case  II. 


38.  When  the  dividend  is  a  compound  quantity,  and  the 
divisor  a  simple  one;  then  each  term  of  the  dividend  must  be 
divided  senaratelv.  and  iha  -refinlfina-  rmonfiViVo  «,:n  v^  i.i-_ 
quotient  required. 


88.  State  the  rule  for  Case  2d. 


DIVISION. 

Ex  1. 

Divide  4iia»+3a5-f-12a«  by  Sa 
42a«4-3«A+12a» 


87 


3a 


-  =  14a  +  i+4a 


Ex.  2. 

Divide  90aV-18a^»+4a'^~2ax  !jy  2a*. 


2<.'a; 


=45aa;'--9.r-i*i*«-|. 


Ex  3. 

Divide  4x*-^2x'-^2x  by  2^:. 

Ex.  4. 
Divide  —  24a«a:V— Sflrary-f-Gzy  by  ■ 
—24a^x^y—Saxy  -f-  Gary 

— 3a:y  ~~" 


-aty. 


Ex.  5. 

Divide  14ai»+7a'6»-21a'i»-f.35a«5  by  7ab 

Hob  "— 


I 


11  \ 


Cask  III. 

titits'  Tn^T•  ^'"^'^^"i  and  divisor  are  io^A  co^j.o.mc?  quan. 

titles     In  this  case,  the  Rule  is,  «To  arrange  both  dividend 

and  divisor  according  to  the  powers  of  the  sime  letter  Wn 
mng  with  the  highest;  then  find  how  often  the  fim  tirS 

he  divisor  IS  contained  in  the  first  term  of  the  dividend  and 
place  the  result  in  the  quotient;  multiply  each  Irm  of  the 
«h;;  n^  andplacJe  the  p?oict  unlTth'e'cor! 

it  fi-om  the^'.  'to  tt^  terms  in  the  dividend,  and  then  subtract 
^c  .^'^.r'fV'  K  *«  the  remainder  bring  down  as  manv  f^rm» 
V.  .uu  uiviaenu  as  wiii  make  its  number  of  terms  equal'to 

AnVr  ^''^''^""^  "°^  '^^^«°'-  '^^^  both  compound  quantities,  what  u. 


:  1 1 


lAej 


O  } 

oO 


ALGEBRA. 


that  of  the  divisor;   and  then  proceed  as  before,  till  all  the 


Ex.  1. 

Divide  a^-Sa'b+Sab'^b^  by  a—b. 


*      * 


In  this  Example,  the  dividend  is  arranged  accordincr  to  the 
Powersoft  the  first  term  of  the  divisor."  HaX  do'ne  this 
we  proceed  by  the  following  steps :—  °  ' 

(1.)  a  is  contained  in  a\  a'  times;   put  this  in  the 


tient 


quo- 


(2.)  Multiply  a-i  by  a\  and  it  gives  a'^a^'b. 
JSJ^  Subtract  a^^a^b  from  a«~3a'6,  and  the  remainder  is 

(4.)  Bring  down  the  next  term  +3a5'. 

(«.)  JVM7/?>/y  and  sM^^rocjf  as  before,  and  the  remaind 


ab\ 


er  IS 


(7.)  Bring  down  the  last  term  —b\      ' 
JS.)  a  is  contained  in  ab\  +5«  times ;   put  tliis  hi  the  qua 


iicnt. 


(9.)  Multiplif  and  subtract  as  before,  and  nothing  remains 


PIVISION. 


89 


•■•f2tfH-« 


la' 


"(n^-Sa', 


Ex.2. 

}-6a<xf  1 00^-7*+ 1 0aV-|-5a««-fz 

"*        3a  V+  7aV4-  5aa;* 
3a V4-  6aV+3aa:* 

*  aV+2a?>a;» 

aV+2aa;*+a;» 


i43<i*'f< 


Ex.3. 


12x*-13x*-34x'4-39a;W+2x'-5x+--f — 
12.i:»-21a;*  \  4c*-7a 


--20^+35^ 


■i  ..  ,fl 


Ex.4. 
32;~6\6a:*— 90    /2x«  +  4a;'+8a:+16 


5\6a:*-9C    /S 


*4-12a;3-9G 
4-12a;^-24x-^ 


*  4.24a;«-96 

4.24a-''-48.t; 

+48a;-9G 


^  I 


r  >l 


li^ 


"  Wlion  there  ia  a  r«mam«fcr,  it  must  be  made  the  numerator  of  a 
Fraction  whose  denominator  is  the  divisor  ;  "this  Fraction  must  then  li6 
placed  in  the  quotient  (with  its  proper  sign),  the  same  as  in  common 
Arithmetic. 


iO 


ALGEBRA. 
Ex.6. 


''■''~')::T::i::-^+^^-'f--^+'' -.+1 


*     +  ar'+ar-l 


Ex.6. 


'"^'felffs+^^^-^^^'-l^+l 


4-  x'^-ix 
*       * 


Ex.  7.  Dh  ide  a*+4a»5+6a=6«+4ad»+i*  by  a+b 

Am,  a'+3a»6+3a6'«+fi». 
£;c.  8.  Divide  «''-5a^^+10aV~10aV+5aa:^-;i^ 

Am.  o*— 2<w.4.;b". 
Ex.  9.  Divide  2bx*^x^^2^^^^  by  5^-4^. 

^ws.  5ar»+4/+3^+2, 
^  10.  i^>videa^+8a3;r+24aV+32a.:3^.ie^.j^ 

-4ws.  a«+6a«ar+12aar«+S^. 


DIVISION.  d\ 

Ex.  11.  Divide  al^—a^  by  a — x. 

Ans.  a*+a*a:+aV+aar*•^x*. 


Ex.  12.  Divide  Gx*+9j^-20x  by  Sj^-Sx. 

Ans,  2aJ'+2x+5- 


5a? 


3a;'-3jr. 


Ex.  13.  Divide  9x«-46.r»+95a^+150x  by  x'-4x-5. 

Ans.  9.J:*— 10ar^+5.»''-30^. 

Ex.  14.  Divide  «*-^a^+««+|x-2  by  ^x-2. 

Ans.  |z='— ^:r»+l. ,, 


PROBLEMS    PRODUCING    SIMPLE     EQUATIONS,    CONTAINING     ONL"? 
ONB    UNKNOWN    QUANTITY. 

Prob.  1.  A  fish  was  caught,  the  tail  of  which  weighed  9 
lbs. ;  his  head  weighed  as  much  as  his  tail  and  half  his  body, 
and  his  body  weighed  as  much  as  his  head  and  tail.  What 
did  the  fish  weigh  1 

Let  2ar= weight  of  the  body  in  lbs. ; 

.-.  9+ar=:  weight  of  tail+|  body = weight  of  head 

But  the  body  weighs  as  much  as  the  head  and  tail  j 

.•.2a:=(9+ar)+9, 

2a:=a:+18; 

.•.a;=18, 

and  .-.  2a;=3r),  the  weight  of  body  in  lbs., 

9+a:=27,  the  weight  of  head  in  lbs., 

and  the  weight  of  fish=36-f  27+9=72  lbs. 

l*ROB.  2.  A  servant  agrssed  to  serve  for  £8  a  year  an<f  a 
livery,  but  left  his  serv  ee  at  the  end  of  7  months,  and  receiver 
only  £2  13s.  4d  and  his  livery  ;  what  was  its  value? 
Let  12j;=the  value  of  the  livery  in  d. 
But  £8= 1920c?.,  and  £2  13s.  4rf.=040d ; 
then,  the  wages  for  12  months=  120?+ 1920 ; 

,*.  the  wages  for  1  month = 


12 


?-f-l()0. 


md  .'.  the  wages  for  7  months = (a; +160)  7, 


I 


i 


4* 


42 


ALGEBRA. 


Cut  the  waces  actually  received  for  7  months=12x.f640' 
.M2a:+640=7a;4.1120; 
.*.  5a?r=480, 
«=96; 
and  .-.  l2x=U52d.=M  16..,  value  of  li^oiy. 
From  the  solutions  of  the  two  preceding  problems  it  will 
1^0  seen,  that  by  assuming/or  the  Lknown%lantTx  will 

A"mption'  hTri^  '''  ?"^^  ^^"^'^"^^'^^  *«  "^^ke  .uch 
m  assumption,   but  a  more  elegant  solution   is  ei^noraU^r 

thereby  obtained      Tl,e  coefficient  of  .  must  b^a^L'S 

proWem    T'"''''  ''  '"   '^'  '^^^^^^^^  ^"^^^^-^   ^nX 

Prod  3.  A  cistern  is  filled. in  20  minutes  by  3  pipes  one 

thJ  Jh  TT'  ^^  ?"""^^«  "'^^^'  ^"d  -»«ther  5  tnons  less 
than  the  third  ;,.r  minute.  The  cistern  holds  820  rilons 
How  much  flows  through  each  pipe  in  a  minute  1        ^ 

Ans.  22,  7,  and  12  gallons,  respectively. 

>^.  finds  hLseif  I160  inSr^^rdti  iT:^^^ 

Ans.  £100. 

Prod  5.  A  met  two  beggars,'B  and  C,  and  having  a  certain 
sum  m  his  pocket,  gave  B  A  of  it  and  C  i  nf  Vhl  ^  cemm 
A  now  hflrl  9ft./  iLa  k  Ju  J  , '  ^^  »  ^*  ^"®  remainder : 
A  now  had  20(1.  left ;  what  had  he  at  first  ?  Ans.  5s. 

£i^^7f\t  \^rT  ^^  ^V  *'^^'^''  *>^^  '^  saddle  worth 
£0^:  If  the  saddle  be  put  on  the  first  horse,  his  value  wiJl  b«v 
come  double  that  of  the  second ;  but  if  it  be  put  on  thelecond: 

lZuf7  ^;\^^^T^  ^"P^'^  *^^^  «f  th«  first,  wtat  is  tl^ 
value  of  each  horse?  ^„,.  ^g^  and  m 

PRon.  7.  A  gamester  at  one  sitting  lost  J  of  his  mm.Av 
and  then  won  18.. ;  at  a  second  he  lost  i  of  the  remS' 
and  then  won  8..,  after  which  he  had  3  guineas  left  How 
much  money  had  he  at  first  ?  ^  ^^ 

Let     15-1!  =  t}lO     rillTy»>»ow     /->r    «U!li: 1_  _      I       1      _  .      - 

having  lost  j  of  hismoneyihe  had  Tof  iro"l2r,^m':i'iS-' 
he  then  won  18...  and  therefore  had  12x+18  in  handTS^ 


i 


ALGEBRAIC  FRACTIONS.  /l.t^ 

.•.(8a;+12)+3=C3, 
8a;+13=:60; 
/.  8a; =48, 
ar=:  6; 
Hence  I5x=90s.=£i  10s, 


CHAPTER   III. 

ON    ALQEBRAIO    FRACTIONS. 

o,i?i;  ^""^  K"les  for  the  management  of  Algebraic  Fractions 
Tt^  TT  ^'  '^'?  ^"  ^'^^^^^^  arithmetic.^  Th^priSes 
?oIIowtg:!!lf  '"^"  "  '^^'  "^^"^^^  -^  established  aTtS 

(1.)  If  the  numerator  of  a  fraction  be  multiplied  or  th^  ^^ 
nominator  divided  by  any  quantity,  the  frac  fon  t Tend^^^^^^ 
so  many  times  greater  in  value.  rendered 

(2.)  If  the  numerator  of  a  fraction  be  divided  or  the  d« 
hJtlu-  ^?  t^Q^nV?^?^^o»'  and  ^denominator  of  a  fraction 


arc  the  foundation  of  tlTScit  K.  Hoicar'''"''  ''*'  ^''""'P'"*  ^^^*^'' 


i 


44 


ALGEBBA. 


ON   THE   REDUCTION   OF    FRACTIONS. 

41.  To  reduce  a  mixed  Quantity  to  an  improper  Fraction, 

Rtle.  "Multiply  the  integral  part  by  the  denominator  -f 
Uie  fraction,  and  to  i\iQ  product  annex  the  numerator  with  iti 
proper  sign;  under  this  sum  place  the  former  denominator, 
luia  Uie  result  is  the  improper  fraction  required." 


Ex.1. 


2x 


Reduce  3a+— ,  to  an  improper  fraction. 

The  integral  part  X  the  denominator  of  the  fraction  +  the 
»wwrator=3aX5a»+2a:=15a'-f-2ar- 
„  15a»4-2a;.     , 

Hence,  — ~-  is  the  fraction  requu-ed.  . 


Ex.2. 


Ax 


Reduce  hx—  ^^  to  an  improper  fraction. 
Here  5j:X6a'=30a«^;   to  this  add  the  numerator  with  its 
proper  sign,  viz.,  -4x;  then  ?2f!^±'  ig  the  fraction  re- 
quired. 


Ex.3. 
Reduce  5a; —-1.  to  an  improper  fraction. 

Here    5a;X7=35ar.      In    adding    the    numerator  2x-3 
with  Its  proper  sign,  it  is  to  be  recollected,  that  the  sign  ~ 

affixed  to  the  fraction  —^  means  that  the  whole  of  that 

fraction  is  to  be  subtracted,  and  consequently  that  the  signs 
of  each  term  of  the  numerator  must  be  changed  when  it  is 
combined  with  35ar;  hence  the  improper  fraction  required  is 
ooa; — 2.g-{-3     33.P  -f-  3 


41.  How  U  A  mixed  quantity  reduced  to  an  improper  fraction  I 


ALGEBRAIC  FRACTIONS. 


45 


Ex.  4.  Reduce  4a6+--  to  an  improper  fraction. 

12a'6-f-2c 


Ans, 


8a 


Ex.  5.  Reduce  36'— —  to  an  improper  fraction. 

156»ar-4a 


^7W. 


Ex.  G.  Reduce  a~a;H -—  to  an  improper  fraction. 


X 


Ana. 


5z 

!.ion 


Am     n 
Ex.  7.  Reduce  3a;' rrr-  to  an  improper  fraction. 


X 


10 


.^n^. 


30a;'— 4ar4-9 
10 


42.  To  rerfwcc  an  *ot        -r  Fraction  to  a  mixed  Quantity. 

Rule.  «  Observe  MHu^^^h  terms  of  the  numerator  are  divisi- 
ble  by  the  denominator  without  a  remainder,  the  quotient 
will  give  the  integral  part;  to  this  annex  (with  their  proper 
signs)  the  remaining  terms  of  the  numerator  with  the  denom- 
inator  under  them,  and  the  result  will  be  the  mixed  quantity 
required."  ^         -^ 

Ex.  1. 

^  a'-^ab-^b* 

i\eauce to  a  mixed  quantity. 

TT       a^-^ab 
^^^  — a — """J"*  ^^  ^^®  integral  part, 

and  —  is  the  fractional  part; 

.*.  a-f-5-f  J.  is  the  mixed  quantity  required. 


48.  W  hat  is  tho  rule  for  reducing  nn  improper  fhwtion  to  a  miied  (^ua^ 


m 


td 


ALGEBRA. 
■     El.  2. 

Keduce ^ to  a  mixed  quantity. 

15a» 


'6a  ^      ^®  '^®  ^^^^S'^^'^  part, 

,  2a:-3c  .     , 
ana  — = —  is  thejraciional  part; 


5a 
2«-3c  . 


.-.  ^H      -^  jg  the  mixed  quantity  required. 
,       Ex.  3.  Reduce  — ^_  to  a  mixed  quantity. 


Ans.  2x— 
Ex.  4.  Reduce  'Jf±^^  ^  a  mixed  quantity. 


5a 
2ir* 


4a 


Ans.  3a+l-_?i. 
4a 


^«s.  10y+3j;-~. 
ar 

43,   To  reduce  Fractions  to  a  common  Denominator, 
Rule.  "  Multiply  each  numerator  into  every  denominator 

tlZ^rt  '^'  "'^  """aerators,  and  all  JZSfr 
fof/ether  for  the  common  denominator." 

Ex.  1. 

p  ■,       2.r  5a;  4a 

neauce  — ,  -,  and  — ,  to  a  common  denominator. 

f^rv-^iv;— TK-    f  rtence  the  free 

5^X^X5-  /5a;    f,  new  numerators ;  I  tions  required  are 


r     Hence  the  free 

4^X3X^1^*  \  "'''  """^^''^^^^^ ;         J  tions  required  are 
4aXrfX6-12a&)  S  jp^^    ^5^    j^^^^ 

5  X 0X5=156  common  denominator;     [  isT'    \Kh>  YbF 
48.  How  aro  fractions  reduced  to  a  common  denominator  f 


ALGEBRAIC  FRACTIO^^S. 
Ex.2. 
tveauce  —g— ,  and  — ,  to  a  common  denominator. 


47 


Hore(2.p.f  1)X4=  8.c+4  )    new  nume. 
^^^^~}2:^___S      rators; 
5X4=20  common  denomi- 
nator ; 

Ex.3. 


'Hence   the  fiac- 
tions  required 
are 

— ^TTT,  and  -— — . 
20  '  20 


Reduce  — j— ,  ~^  and  — ,  to  a  common  denominator. 
Here  5a;X  Sx2a:=30j:»  >  .*.  the  new  frac      30ar* 


•  '.1 


{a^x)     (a+x)x2x=2a'x-2x^  tions  are     6^*6^ 

1      X(a+:r)x3  =Sa    +Sx    ha'x-2.^        ,     3a4  3.r 


(a-i-x)  X3  X2x        ==Qax  +Gx^j  QaTTO?^  ^"^  CllJ+e?' 
Ex.  4.  Reduce  — ,  g^,and  -^,  to  a  common  denominator. 

"":''  &  l5^'  ^"^  157- 
Ex.  5.  Reduce  --^,  and  — g~  to  a  common  denommator. 

Ar.j.  —- — ,  and 


3a: 


3a; 


Ex.  0. 


4a;'-|~2a;    3a;'  2a; 

Reduce  — - — ,  — ,  and  -,  to  a  common  denominator. 

J       4Sabx^-{-24abx  45ia;»       ,  40aa; 
C0a6         '  mab  '  ^'"^  QQab' 

Ex.7, 
o  ,       7.r'~l        ,  4i:*— ar+2 


Ze 


aa" 


An,    '±'.f=K  .ni?^Zl-:+*^ 


4a*x 


,  and 


4a'at 


I     : 


18 


ALOEBKA. 


44.  To  reduce  a  fraction  to  its  lowest  terms. 
Rule.  "  Observe  what  quantity  will  divide  all  the  terms 
both  of  the  numerator  and  denominator  without  a  remainder; 
Divide  them  by  this  quantity,  and  the  fraction  is  reduced  to 
^ts  lowest  terms." 

Ex.  1. 


Reduce 


14a;»+7qar-f21a;» 


to  its  lowest  terms. 


The  coefficient  of  every  term  of  the  numerator  and  denorai- 
nator  of  this  fraction  is  divisible  by  7,  and  the  letter  x  also 
enters  into  every  term  ;  therefore  7x  will  divide  both  numa- 
rator  and  denommator  without  a  remainder. 


Now 


14a:»4-7aar+21a;« 


iX 


=2a;«+a+ai?, 


and  ---=5a;; 

7x  * 


Hence,  the  fraction  in  its  lowest  terms  is  ^^'"^-JJf. 

5x 


Reduce 


Ex.2. 

20a5c~5a-+10ac 
5a^c 


to  its  lowest  term. 


Here  the  quantity  which  divides  both  numerator  and  de- 
nominator  without  a  remainder  is  5a ;  the  fraction  therefore 

in  its  lowest  terms  is  — ^~"+  f 


ac 
Ex.  3. 

Reduce  — — -.  to  its  lov/est  terms. 
(r—o* 

Here  a—b  will  divide  both  numerator  and  dcnominatoi, 
for  by  Ex.  2     Case  III.   page  27.  a^—b''=(a-{b){a—b)i 

hence  — -— •  is  the  fraction  in  its  lowest  terms. 

tt"f~0 


Ex.  4.  Reduco  -—  to  its  lowest  terms. 


2jr 

Ans.  — , 


44.  Show  how  fractions  are  reduced  to  their  lowest  tenon. 


\ 


ALOEBKAIO  FKACTIONS. 


49 


Ex.  5.   Reduce  — -^  to  its  lovrest  terms. 


Ans.  ~. 

(it 


Fx  6.  Reduce ^-^         ^.  to  its  lowest  terms. 

2y--3jry 


Ans. 


X 


Ex  7.  Reduce r^-^—  to  its  lowest  terms; 


Vtx^ 


Ans. 


3j:«--ar4-2 


a^ 


ON    THE    ADDITION,    SUBTRACTION.    MULTIPLICATION,     AND 
DIVISION,    OF   FRACTIONS. 

45.  To  add  Fractions  together. 

Rule.  "Reduce  the  fractions  to  p.  common  denominator 
and  then  add  their  numerators  together;  brin<T  the  resulo 
mg  fraction  to  its  lowest  terms,  and  it  wiU  be  the  sum  re- 
quired." 

Ex.  1. 

"5'  T'         3'  ^^g^t^er. 

2a;X5x3=30a;  I      63a:4-30a:-|-35aj      128^ 

icX5x7=35a;  (  •*• jQg =  -y^  is  the   fraction 

5X7X3=105J  "^        required. 

Jix.  ^.  Add  -,  — ,  ana  — ,  together. 


«X34x4a=12a''61 
2aX6X4a=  8a«6 
56X6X36=156' 


.-.  lSa^^+8«'6-fl568  __  20a«6+15^>' 

12a6»  ~       12«6^-" 

6x36x4a=i2a6'J    =  (dividing  by  6)  ^^^^f  is   the 

sum  required. 
46.  State  the  rule  for  addiiiji  fraction«i. 


i 


lis 


■ 

i; 

'I 


50 


ALGEBRA. 


Ex.3.Add?^,?^i, 
5     '     2x   ' 

i2x-\-3)x2xx7=2Sx'  +42z 
(3a;^l)x5   X7=105a;~35 
4arx5x2ar=r40a;^ 


5x2a;x7=70a: 


4x 
and  — ,  together. 

^   28a;»-f42g-fl05ar— 85-i»40r» 
70a- 
68a:''+147ar-35    , 


70a? 


s    the 


Ex.  4.    Add  --,  ~,  and  ~,  together. 

Ans. 
ir-x.  o.   Add  2^'  "5'  ^^*^  7"»  together. 

Ans. 


sum  required. 


934a; 
693* 


105a'-f28a'6+3C6« 


Ex.  C.    Add  ?^i  lf+?,  and  |,  together. 


70a6 


Ans, 


Ex.  7.    Add 


5a«+6 


169a;+77 
105     • 


4««4-2^ 

Ans. 


36    '  *^"^  "sT"'  ^^g^^h®''- 

SToM-m 
156"     ' 


Ex.  8    Add  -1^,  and  — ,  together. 

.       4«»— 7a?— 3 
Atis,  - 


Etj,  i>.    Add  --— ,  and  — —   together 


6« 


Ans. 
Ex.  10.  Add  — -..  and  — r- .  v»o-PtliAr 


2a?« 


tt— 0 


a  +  o 


^n*. 


^M-26« 
a»-6» 


ALGEBRAIC  FK ACTIONS. 


5J 


40.  To  Subtract  Fractional  Quan.ities. 
RiTLE.   "Reduce  the  fractions  to  a  common  denominator  • 
and  then  subtract  the  numerators  from  each  other,  and  uudw 
the  difference  write  the  common  denominator." 


3xXl5=45a; 
UxX  5=70ar 

5  Xi5~75 


Ex.  1. 

Subtract  ^  from  — . 
5  15 

•*•       75       —  if^  ~  3  ^^  ^^®  difference 


required. 


Ex.2. 


Subtract  ?f±3  from  ^"^^^ 


3  7 

(2.r+nx7=14ar+7)        15ar+6--14.r-7      ar-^1 

(5a?+2)x3z=15ar+6  \    •*• ^l ~"  =  "sf  ^^  ^^ 

3x7=21         )    fraction  required. 

Ex.  3. 

From  i5|r2  subtract  ^1, 

(10u:-9)x7=70a;-63 )        70x-63-24a:+40      4Gr-^3 
(.3^5[x8=24^-40[   .-. -^-±-=:15l^ 

"     ' "^     '"  is  the  fraction  required. 

Ex.  4. 


8  X7=56 


From r  subtract . 

a—b  a+b 


(a-i)  (a+6)=a«~"65  J  ^^  j^  ^j^e  fraction  required. 


( 


Ex.  .5.    Subtranft  If  frnm  ?? 
5   10' 


"'^•lo- 


4«.  Give  the  rule  for  subtracting  fractious. 


>n 


vt: 


il 


62 


ALGEBRA. 


'     I 


Ex.  6.  Subtract  5^  from  ^i^ 


fix.   7.  Subtract --±1  from  If 
x+l  5* 

Ex."  8.  Subtract  ~-=:3  from  1^ 

Ex.   9.  Subtract  • — -  from    ^ 

a-\-b  a~b' 

Ex.10.  Subtract --Hl  from - 

8  7* 


^,«.  lE^iZ 


Ans. 
Ans. 


28 

5ar-{-5 
4x'+3 

l_l£+49 
66     • 

8 


*    47.  To  Multiply  Fractional  Quantities. 

Rule    "Multiply  their  numerators  together  for  a  nrw 

numerator,  and  tW  denominators  together  for  a  new  d^ 

L'rmsT'  '"'  '^'"^'  *'^  ^""^^^"S  ^-<^^--  to  its  Wt" 

Ex.  1. 
Multiply  ^  by  1^. 

7  x9  =63  f  •*•  the  fraction  requireu  is -- 
'  ^63 

Ex.2. 

Multiply  1^+1  b/' 


3 


Here 

C4«-f-l)x6«=24z»-|-6 
and 

3x7  =21 


r      24ar«-f  Gar       , 
la;  I  *  *       21 —  ~  ^dividing  the  nu 

1  JTierator  and  denominator  bv  3'' 
arM-2ar  *      ' 

ij; —  IS  the  fraction  required. 


(« 


47.  State  the  rule  for  the  multiplication  of  fractio:, 


M, 


ALQEbRAIC  FRACTIONS.  53 

Ex.  3. 

00       "  a-\-b 
By  Ex.  2.    Case  III.    page  27,   (a^-i»)x3a'=(a-t  5) 
(a-i)x3a«;   hence   the  product  is   3a'X  (a+^)(a-j)^_ 

56x(a+6) 

(dividing  the  numeratorand  denominator  by  a+6)  ^^'^C^-^) 
3a»-3a»6  ^^ 


56 


Ex.4. 

Multiply  54^%y      '^« 


14 

Here 

(3a:«-5a:)  X  7a=21aa;'-35aa; 
and 

(2««-3a:)  X  14=28a:»-42.r 


2a;''-3ar* 
^         21a.r«-35aar       , 

the  numerator  and  denomi- 
nator  by  7x)  ^=|  is  th. 


J    fraction  required. 


2x 


Sx 


Ex  5    Multiply  -5.   by  ~ 
Ex.  6.   Multiply  ?^i:f  by       ^^ 


2a:*-4aj* 


Ex.  7.   Multiply  ~  by  —1'. 
Ex.  8.  Multiply  --5fL  by  ^^^-30 


2a: 


Ans. 
Ans. 

Ans, 
Ans. 


Ix-^' 

3ar-~l 
ar-2" 

4 

2' 


I-' 


^ 


48.   0/j  the  Dlvmon  of  Fractions. 
tion  "^'''*  "'^'*^^^^  ^"®  divisor,  and  proceed  as  in  Multiplira 


48.  Enunciate  the  rule  for  division  of  iVact^ona. 


»jira)ll>fcawiLMWtoii*»w»Mainwn«M'iii;/.;:*'TT^ 


■--'—y  jwjtwi  ■- 


M 


ALGEBRA. 


II 


Ex.  1. 
Divide  —  b/  -. 

Inter t  the  divisjr,  and  it  becomes  ~ ;   hence  —  x  — 

-  JqJ  ~  "s  ^*     ^'^'"^  ^"®  numerator  and  denominator  by  Qx) 
s  tlie  fraction  required. 


Ex.2. 


Divide  ^^^  by  ^^^-^ 


14ar-3  25' 

X 


6 


5         -^      25     • 

(143:-3)x5  _  703?- 15 

lOr -4  10^-4      ~  10x^=^4* 


Ex.  3. 

Divide  ^^'  by  lli±. 

3a^-5&«_5  X  (a+&)(«--^)  f.  5x(a-f-^^)(a~&)^. 

2a      ~  2a 

4a-4-46_4x(a  +  ^) 
66     "        66        ' 


66 


JS 


Ex.  4.  Divide  ^  by  ^. 


2a  ^4x(a+6) 

i  _306x(g-6)     15a6--15^"  . 
~"         8a  "^        4a 

the  fraction  required. 

\ 

^'^^-  6l- 


Ex.  5.  Divide  ~±2  by  ?^i. 
o  5a; 


Ex.  0.  Divide  ~  by  ^. 


.4n5. 


^n«. 


10.g 
3  • 

4ar--12 
5    "' 


_      _    _  9r«— 3rr         rs* 

fix.  /.  jnvide  — :; —  by  — . 


Ans, 


n«      o 


SIMPLE   EQUATIONS. 


6u- 


ON     TIIK    SOLUTIOif    OF    SIMPLE     EQUATIONS,    CONTAININO    ONLf 
ONE    UNKNOWN    QUANTITV. 

Rule  III. 
Inc?!;.^"-!*^"^..^'?"  "'^>'  ^^  ^^^^^^^  «^  fractions  by  mullinlv.. 
SsTnVutlt^  ^^"^'^"  '-  ^'^  denominators^of  thS 

eac^'Llde  Tth«''"  ""?  ^e  cloared  of  fractions  by  multiplying^ 

s^iititf  trSiot^^^  '^"  ^^^'^-^'^  -/.>/.y  thf : 

This  Rule  U  derived  from  the  axiom  rd\    fi,of   ,•<•  i 

2"rjf^^ "^"^^!p"^<i  by  thetr^j^;.*  ^i^ :s 

quantities),  the  products  arising  wUl  be  e  laal  ^    ^      ^  ^ 

Ex.  1.  Let  1=6. 
Multipli^  each  side  of  the  equation  by  J>,  then  (since  tha- 
multiplication  of  the  fraction  |  by  3  just  takes  away  the  de. 
nominator  and  leaves  x  for  the  product)  we  have 

ar=6x3=18. 

Ex.  2.  Let  |4-|=7. 
Multiplj^  each  side  of  the  equation  by  2,  and  we  have 

^Ag,m,«„/.>/yeach^,HerfJhi,  equation  by  5.^        „. 

7a:=:70'; 
Var=10. 

Ex.3.  Let^  +  -=i3__f 

Multiply  each  side  by  2,  then    a:4.- =2G-?? 

3  4 


Mullinlv  oanh  oJ/la  !»».  O j      c!._  .   ^  __        


6^ 
4' 


Multiply  each  side  by  4,  and  12x-\-8x=zm^^(kc. 


S8 


ALGEBRA. 


By  traiispt  iition,  12j;-f  8a;+-G.r=::312, 

26a:=312; 
.-.     ar=  12. 

Tliis  example  might  have  been  solved  more  simply,  Ly  mul 
rL'])l3ing  each  side  of  the  equation  by  the  least  common  multi 
pie  of  the  numbers  2,  3,  4,  which  is  12. 


Multiply  each  side  by  12,  -7; — |- 


2 


:156-1?A 


or,  Qx-^4x=zl56—Sx. 
By  transposition,  0a;-f-4a:4-3a?=156, 
*  13x=156; 

,\x=  12. 


Ex. 

4. 

Let 

2x    X 
3  "^4" 

:22. 

Ex. 

5. 

Let 

7x 
4" 

5x 
a' 

55 
-6' 

Ex.0.  Let  ^.+5=31-1. 
<«    «>  o 

Ex.  7.  Let  ^-~+f =44. 
5      ()     2 


Ans.  a: =24. 
Ans.  ar=:10. 
Ans.  x=SO. 

Ans.  ar=GO. 


60.  In  the  application  of  the  Rules  to  the  solution  of  simple 
i.juations  in  geocral  containing  only  one  unknown  quantity,  il 
will  be  proper  to  observe  the  following  method. 

(!)  To  clear  thj  equation  of  fractions  by  Rule  III. 

(2.)  To  .  ol'ect  the  unknown  quantities  on  one  side  of  the 
-Cfjuullon,  fjid  ike  known  on  tl  e  other,  by  Uule  II. 

(3.)  j'o  find  the  value  of  the  unknown  quantity  by  di- 
viding TAcU  g/iie  of  the  equation  by  its  coefficient,  lis  in 
Rule  I. 


60   F.)  I  A  -'    ■:  tie  thrco  Btups  Vr  which  0  B'mpie  equation  containing 
only  oivj  •.'•-'/>  -vn  iiaantity  rany  be  BolVoU. 


SIMPLE  EQUATIONS.  57 

Ex.  1. 

l^uvi  'Jie  value  of  a:  in  the  equation  —  -  l=f-|-l?. 

7  5     5' 

Multiply  by  7,  then   3^-fV=-+-. 

Multiply  by  5,  then  15j;4-35=7ar-j-91. 
'X)llect  the  unknown  quantities  on  ) 
one  side,  and  the  known  on.  the  [  15.r— 7;i:=9I— 35 
other;  1  '^' 

or  8z:i=5G. 

Divide  by  the  coefficient  of  x,  ar3=— — 7 

'        8  ~  * 

Ex.  2. 

Find  the  value  of  a:  in  the  equation  ^i?-  1=:2— - 

5  7* 

Multiply  by  5,  then  ar+  3-  5  =  10-—- 

7  * 

Multiply  by  7,  then  7x4-21  ~25=70-5ar. 
Lollect  the  unknown,  quantities  ) 

on  ont   side,  and  the  known  >  7ar-f-5ar=70— 21-f-35. 
'.•n  the  other  ;  \ 

orl2ar=84; 

••''■-l2~^- 
Ex.  3. 

Find  the  value  of  x  in  the  equation 


x—\ 


2:e~2 
~5 


-24. 


Multiply  by  the  least)  .^       r    ,  ^     ,^ 
(vmmon  multiple  (10),      f  40«-5ar+5=10.r4-4.r-4+240. 
Uy  transposition,  AQx-^bx—lOx—  4a'=240— 4— 6. 

or40a;-19.r=231, 
i.  e.  21a:=231 ; 

231     ,, 


IWA 


As  the  Jirst  step  in  this  Example  involves  the  case  "  where 
the  sign  -  stands  before  a  fraction,"  when  the  numerator  of 


./  -M^^mmm^ 


^ 


ALGEBRA. 


! 


that  fraction  is  brought  down  into  the  same  line  with  40ar  the 
signs  of  both  its  terms  must  be  changed,  for  the  reasons  as. 
Signed  m  Ex  3,  page 44;  and  we  therefore  mai«e  it  --5.r-f  5 
and  not  5af— 5.  ' 

Ex.  4. 

Find  the  value  of  a:  in  the  equation  2a?— -  -f  1  =5ar~2. 

Multiply  by  2,  then  4ar--a;+2= 10a?— 4. 

By  transposition,  4+2=10a:--4a;-f  ar, 
or  6=  7x; 
6 

6 

t       or  x=-. 

Ex.  5. 

What  is  the  value  of  a;  in  the  equation  3az-{-2bx=Sc+a'i 

Here  Sax-{-2bx={Sa+2b)xx', 

.•.{Sa+2b)xx=:Sc+a. 
Divide  each  side  of  the  equation  by  Sa+2b,  which  is  iha 


coefficient  of  a:;  then  a;==?^il^ 


'Sa-i-2b' 
Ex.  6. 
Find  the  value  of  a?  in  the  equation  Sbx+a=2ax+4f, 
Bring  the  unknown  quantities  to  one  side  of  the  equati(A.   ui/^ 
the  kno%on  to  the  other  ;  then, 

36a;-— 2aa;=4c--a;  ' 

but  36a;— 2aar=(36--2a)  Xar ; 
.'.  (36-2a)a;=4c-a. 

Divide  by  36-2a,  and  «=|^. 

Ex.7. 

Find  the  value  of  a;  in  the  equation  6a;+.r=2a;+8a 
Transjose  2a;,  then  bxA^x    2a?=i3a, 

or  bx—  a;=:3a ; 
but  6a?—  a;=(6— l);r; 
.  (6—1^  a;=3a. 


and  x-rz 


m* 


w^tmmmmi^m-'r'nk 


ssasfi 


Ex.   8    a:4-4.-~ii 


SIMPLE  EQUATIOXS.  ^j^ 

Ans.  xz=.Q. 


"        5^4^3    2^     • 

^         2^3    4~2' 
-E:x.\12.  3^+i     *+5 


9~"~F'* 


—  Ex.  ^3.  -f«5=:,2^_2^ 

-  Ex.|l4.   C^-^-.9=.5:r. 

"Ex.15.   2x~^^+i5^1?i±20 
3  6 

-^    Ex.lG.  ^Z:?+^-2C-^-^ 
2    ^S"-"^^      ~F"- 


7 

-4?w.  ar=rl4. 
-4««,  ar=3G. 
Ans.  «=12 
-4n*.  a?=:18. 


2ar--l 


Ex.  17.  5^~ffll^4.1-3^.fi:2.^      . 

3     ^^--«>*i     ^ — f-7.  Am.  z=8. 


Hoc.  18.    2az-{-b='6cxi-ia, 


Am.  z=^. 


Ex.  19. 

Multiply  by  15,  45x-.60-2S^4'm^7oj_A^     a 

45ar-28ar^4r=72l4+ 60-30 
13i;=:92;  ' 


"; 


t^ 


""^B^w^!*". 


eo 


ALGEBRA. 

Ex.  20. 
4«-y    7a;-29_8ar+10 


18 


;  fmdx. 


126ar~522 

Multiply  by  5;r-12,  126:^- 522=65.^-150 

126a?-65;r=522-150! 
61«=366; 
.-.  ar=6. 

*     Ex.  21. 

Given  ~? L  -      1 

«~1     a:-|-7"~7"(«-l)'  *<>  fi»«*  * 

Mult,  by  7(x-l),  7-.lifei:l)^i^ 

»*  "f*  7 

6=:iiferJ) 

a;-H7     • 
Divide  by  2,        sJltnl} 

3ar4.21=7:c-7, 
7«— 3ar=2l4-7, 
4aJ=28 ; 
.-.  a?=7. 

Ex.22. 

Let  ?^i4-2ir?-.I«^+J5  .  2J  •      ^ 
14   ^6«+2~'     28 r-^;find« 

Mult  by  83,  ,6.+ 10+l^r|f=i«,+ ,5^„] 

196;r— 84 

-==14 

196ar--84=84ar4.28.     * 


~^-\ 


V. 


V. 


0, 

the 


iJgfflSWft*- 


SIMPLE  EQUATIONS. 


63 


V. 


4  3     -^i 9~-- 

Ex.  24.  ?f±?2_l^-12,  ar 
36     ~5^34^-4' 

Ex.  25.  ?2^±^  .  5f+20__4.r    80   ' 
25      ■^9jr-16~"5'^25* 


V 


Ant.  0. 
^n*.  8. 

An9,  4. 

Ex,  26.  ?^S_i2fr5  ,  f  _7«    ^+16 

9  17a;-32"*"3~12 36~*        ^"*-  ^• 

Ex.27.  4(5.-3)-G4(3~.)->3(12.~4)=96.  Ans.  0. 
Ex.28.10(.-f|)^0.(l^j).23.  ^,,^ 


Ex.  29.  -?±?f4.5£±?f_,.^  48 


^««.  3. 


t:H-3 

PROBLEMS. 

Let  a;=:the  number  required  • 
then  ;.+ loathe  number,  Wth  10  added  to  it. 
Now  Jths  of  (ar+10)==?(a:^-10)=l(f±i£)_3£^-30, 
But,  by  the  question,  Jths  of  (ar+ 10) =66  •  ^ 

•    Hence,  5f+22^ee.' 
Multiply  by  5,  then  3af+30=330 ; 

.•.3^=330-30=300;  or.r=?^=ioo. 

the  quotient  shall  be  20?  ^      '      ^  ""^^  '''''  ^'^'^^'^  V  » 

Let  ar=the  number  required  ; 

tjien  6^=the  number  multiplied  hv  fi. 

Ox-t-io=ihe  product  increased  by'is/ 

6.r-}-18      ,  •'^     °» 

ana  — = — -ih^it  sum  divided  bv  9 


^ 


9 


6 


: 


1! 

J 


62 


ALGEBRA. 


Hence,  by  the  qu^  ion,  ^i+l^^g^^ 
Multip]j^by9,then6:r+18=:l80 


1  ROD.  3.  A  no«!<-  io  i*k  •     1  6  ~"     * 

^et  out  of  tho  C  fe  tSf '  ^^^j^  -ate,^  and  13 

I^et  .^.length  ofThe1>o:t":  j.^^^  ^^^^«  P-t  ? 

then  |=the  part  ofit  in  the  earth. 
3ar 

y=^the  part  ofit  in  the  water, 

Butparur'ii^nr'^'""'''^^^^^^*^^- 
whole  post ;  '  ^  ^  ■  '      "  ^al'^  +  part  out  of  water  = 


I    -^ 


(t)    '• 


^■2        s=  X. 


Multiply  b.y5,  then. +  '•-_,    65^5^. 
Multiply  by7,  then  7^+ ,5^1:455^35; 

Prob  4    Afv  .  13  -"^^  ^^"gtli  of  post  in  ft. 

had^«-  f  V  ^fte^  paying  away  kh  and  ith  nf 
ftad  i^5  left  ,n  my  purse.     WhJ  rnnnlv^v-^^  r  "'^  ^«««>^  » 
Let  ^^.>oney  in  purse  a?  fim ;  ^  ''  ^'^'^  ^ 

then  j+-=money  paid  away. 

(4+7J       =     85, 

Multiply  by  4,  then  4z~.x-^^S40' 

Multiply  by  7,  then  28.-7.^41=2380. 

.-.  17.=2380: 


X 


■=27. 

,  and  13 

t? 


^aterss 


iinft. 

ney,  \ 
? 


nirg; 


SIMPLE   EQUATIONS. 


98 

^rol^ll^fZsl^^^^  «<Jd  20,  and 

suDtract  1^,  the  reraaaider  shall  be  10  ? 

Ith'^W  ^"^^  """'"'^  "  "'»'  ^h"^  id  part  cxceeda  U, 

^«*.  540, 

thi«  difference  divided  bvTttn.-^''"^!^^^  ^^^ 
arc  the  numbers?  ^    '     '  ^""'^'^"^  7^^^  ^«  «.     What 

4»ff.  21  and  16. 

mainder  will  be  5.     What  a,!  thenJllt  ^'''""'  ""^  ^" 
p  ^w*.  35  and  41 

the  sto„4Jl!;Ztll\S  ^  '^'  """""■■  ^""^""^  "' 
P„„     > ,     »  ^"''-  •*"•  ^'*=  "'"J  13- 

mcome.  t.  l  oi  it  ^^m«  ij^s.     p,e(|uired  his 

^/<*.  £150. 
Prob.  12.  A  gamester  at  one  aittina  lost  Ifh  nf  k.-c 

guineas  left.     wStr„e^  fa'^R  fit,'  ""'^'^t:  '^^  ' 

lo  the  same  quantity  7    '  "''•''  ""  ''«  «q«»l 

^         •'^'  -^'"-  18, 23,  10, 40. 

»/«TpeHb^bT^t"^^^*"  Jy■."f.^«^  «~...<'-  worth 
-u™  he  take  ,»  fi.m  a  oheVtV To4  ^:.^^CZS, 

Ana,  33  at  IS*.  M. 


1 


''1  at    9*.  6ci 


i# 


i  I 


!i:i 


64 


ALGEBKA. 


<; 


(( 


(( 


1  uoB.  15.  Three  persons,  A,  B,  and  C,  can  separately  reao 
ft  i.eJd  of  com  m  4,8,  and  12  days  respectively,  lii  how 
luanj  days  can  they  conjointly  reap  the  field  ? 

I     [.et  ;r  =  No.  of  days  required  by  them  to  reap  the  field; 
then  It  1  represent  the  work,  or  the  reaping  of  the  field, 
■J^=the  part  reaped  by  A  in  1  day. 

j_    «        ;;  «  •'    g  ^» 

T^=    "       "  "  C 

.'.i+l-\-j'^=z  «      «        «     all  three 
But  the  part  reaped  by  all  three  in  1  day  multiplied  by  the 
number  of  days  they  took  to  reap  the  field,  is  equal  to  the 
whole  work,  or  1 ; 

•••(i+i+A)^=l: 
Clearing  of  fractions  by  multiplying  by  24, 

(6+3+3)  a;=24, 
llar=24; 
i  .'.  x=2^^  days. 

Prob.  16.  a  man  and  his  wife  usually  drank  a  cask  of 
*»eer  in  10  days,  but  when  the  man  was  absent  it  lasted  the 
wife  30  days  ;  how  long  would  the  man  alone  take  to  drink 
'^'  Ans.  15  days. 

Prob.  17.  A  cistern  has  3  pipes,  two  of  which  will  fill  it  in 
3  and  4  hours  respectively,  and  the  third  will  empty  it  in  C 
hours  ;  in  what  time  will  the  cistern  be  full,  if  they  be  all  set 
a-running  at  once?  jins.  2h.  24m. 

Prob.  18.  A  person  bought  oranges  at  20c?.  per  dozen ;   if 
he  had  bought  6  more  for  the  same  money,  they  would  have 
cost  4c/.  a  dozen  less.     How  many  did  he  buy  ? 
Let  ar=rthe  number  of  oranges ; 
thena;+6=  «        "       "        "       at  4d  less  per  dozen. 

Price  of  each  orange  in  lstcase=fft=4d'. 
and  «    "     *'        «      «  2d    "   =11 =k 


,*,  the  cost  of  the  oranges =- 


5x 


3* 


■r 


But  we  have  also 


inv  cifSi  ui  tne  orauaes=*  ia;-f-0}. 
Two  independent  vrlues  have  therefore  been  obtained  foi 

9 


f\\ 


SIMPLE   EQUATIONS. 


65 


the  cost  of  the  oranges ;  these  values  must  necessarily  be  coual 
to  each  other ;  •'  ^ 


to  each  other ; 


5^ 

.••3=1(^+6). 


Multiplying  each  side  of  the  equation  by  3, 

p         ...  •'•  «=24,  the  No.  of  oranges. 

rROB.  19.  A  market-woman  bought  a  certain  number  of 
apples  at  two  a  penny,  and  as  many  at  three  a  penny,  and 
sold  them  at  the  rate  of  five  for  twopence ;  after  which  she 
found  that  instead  of  making  her  money  again  as  she  expected, 
she  lost  fourpence  by  the  whole  business.  How  much  money 
had  she  laid  out]  ^  Ans.Ss.4d'' 

Prob.  20.  A  person  rows  from  Cambridge  to  Ely  a  di«^ 
tance  of  20  miles,  and  back  again,  in  10  hours,  the  stream 
Howmg  uniformly  in  the  same  direction  all  the  time  :  and  he 
mds  that  he  can  row  2  miles  against  the  stream  in  the  same 
time  that  he  rows  3  with  it.  Find  the  time  of  his  ffoin<r  and 
returning.  *'      '° 

Let  3a;=:No.  of  miles  rowed  per  hour  with  the  stream  • 
,'.2x===  "     «      «         «         «      «     against        "      ' 
^ow  the  distance  divided  by  the  rate  per  hour  gives  the  time ; 

**'  ¥x~  of  hours  in  going  down  the  river, 


3x 

J  20 
and  — -  = 

2x 


(C 


(( 


It 


u 


coming  up 


(( 


But  the  whole  time  in  going  and  returning  is  10  hours  • 

20     20 
"Fx'^2:v^' 

Dividingby  10,  .-^+1=1. 

r>.5         X 

Multiplying  each  term  of  the  equation  by  3x, 
2+3=3^; 

.'.x=-=ll 

and  .-.  32- =5,  miles  per  hour  down 
-  .t     ..       ._       .       .  2^ 

,-.  ine  :ime  m  gomg  down  rhe  river=:— =4  ^ours 

5 
ne  of  returnin 


on 


wquciitly  the 


0* 


:  10-4 =6  hours. 


0li 

4 


' .  ■  'Wl^^  «?E»«^ 


m 


ALGEBRA. 


Prod.  21.  A  lady  bought  a  hive  of  bees,  and  found  that  the 
pr,ce  came  to  2..  more  than  |-ths  and  Jth  W  the  price      fI^^ 

Ans.  £2. 
Prob.  22.  A  hare,  50  leaps  U^c<.  ..     .oyhound  takes  4 
eaps  for  the  greyhound's 3 ;  but  W..,  of  tUe grVho^id's leaps 
are  equal  to  three  of  the  k.  .'..     How  mafy  iapTwIl    ?he 
greyhound  take  to  catch  the  hare  ?  ^ 

Let  X  be  the  No.  of  ieapn  taken  by  the  greyhound  ; 
then  -  will  be  the  corresponding  number  taJ,  a  by  the  hare. 
Let  1  represent  the  space  covered  by  the  hare  hi  1  leap; 

then  -  «         4i       «4         u  u  ,        , 

2  "  .        "      greyhound  « 

.4a:  4x     , 

. .  g  X 1  or  y  wil'  be  the  whole  space  passed  over  by  the 

hare  before  she  is  take. ;  and  :.  x  |  or  |  .vill  be  the  space 
passed  over  in  the  corresponding  time  by  the  erevhound 
Now,  by  tho  problem,  the  difference  betwo'n  ?l7spaces 
Srps^;^"''  ^^"  '^  the  greyhound  and  harl  LTx" 
.3a:    4x     _ 
• '  2  -  3  =^^^' 
Oar— 8a;=:300; 
.'.  ar=300  leaps. 


ON    THE   SOLUTION   OF   SIMPLE    EQ.        ^ONb,    C0NTAI^.VO   TWO 
OR   MORE    UNKNOWN    QUANTITIES. 

51.  For  the  solution  of  equr  'om.  coutaining  tw  ,  or  more 
u  known  quantities,  as  many  independent  equations  arrrl 
quired  as  there  are  unknown  quanUties.     The  two       It;.m 
necessary  for  the  solution  of  the  case  when  t^vn  „nl  n 
«es  are  concerned,  may  be  ^sI^V  ^  ^  ^"^ 

ax-{-by:szc 
a'x-\-b'y=:c\ 
Where  a,  6,  c,  a\  b\  c>,  represent  known  quantities,  and  z,  y, 


SIMPLE   EQUATIONS. 


67 


thQ  unknown  quantities  whose  values  are  to  be  found  in  terms 
ot  these  known  quantities. 

There  are  three  different  methods  by  which  the  value  of 
fMic  of  the  unknown  quantities  may  be  deterniin  d. 


J^ 


ml 


FIRST   METHOD. 

^i*^'^  t^e  value  of  one  of  the  unknown  quantities  in  tenns 
ol  the  other,  and  the  known  quantities  by  the  ruJ.  s  already 
given.  Find  the  value  of  the  same  unknown  quantity  from 
the  sofond  equation. 

Put  these  two  values  equal  to  each  other ;  and  we  shall 
then  have  a  simple  equation,  con^Mning  only  one  unknown 
quantity,  an  hich  may  be  solved  as  before 


•gx  [■  to  find  X  and  y. 
a) 


Ex.  1.      Given  x-\-y=S  -  ...  (1) 
x—yz=4:  -  - 
From  (1)     y=S~x  -  - 
"     (2)    y=.-4 
Putt'ng  these  two  values  equal  to  each  other,  we  get 
ar~4=.8--ar, 
2^=12; 

X=z:6. 

By  (      y=8-a:='8--6=2. 

Ex.2.  Let       4y=.16 (1) 

4.  ,    y~M (2) 

From  equation  (1),  v     have  a;=16— 4y. 
(2)    «     «« 34— y 


16 


in 


(( 


u 


«  «         XT-' 


Hence  by  the  rulo^         ^=10 -'4y, 

34-y=6     -16y, 
15y=30j 

. .  y=2. 

It  has  already  bee     shown  that  ar=:I«-_4y=/'since  ?/  -2' 
and .-.  4y=8)  16-8     8.  ^  ^       ' 


/d 


'/ 


51.  For  tlio  soiution  of  equations  containing 
titios,  how  uany  independent  equations  arc 
niHliod  ol'auiution. 


•-  more  unkno     1  qaan 


S^^asawr**- 


1 


Q8 


Ex.  3I 

Ex.4.! 


ALGEDRA. 


\ 


5z+3y=38  ) 
4x—  y=10J" 

2^-~3v  =  — 1  ) 
3x-2y=6      f 


Ans.   1^=4 


SECOND   METHOD. 


1^1 


j  From  either  of  the  equations  find  liie  value  of  one  of  (h,. 
unknown  quantities  in  terms  of  the  other  and  the  known  quau. 
Jties,  and  for  the  same  unknt  wn  quantity  substitute  this  value 
(n  the  other  equatior,  and  there  will  arise  an  equation  which 
Jontams  only  one  unknowi  quantity.  This  equation  can  be 
solved  by  the  rules  already  laid  down, 
I   Ex.  1.  i/-'X=2  .  - 

'       From  (1)        y=2+x.        (a) 
This  value  of  ?/  being  substituted  in  (2),  eivet 

2a!r=6; 

And  by  (a)  y=2+a;=24-3=5- 

Ex.  2.  x+2  ,  ^ 

-3-+8y-31  (1) 

~  +  10a:=192         (2) 


Gearing  equation  (1)  of  fractions, 

a;+2+24y=93,  or  a;-f-24y=91 
Uearing  equation  (2)  effractions, 

y+54-40a;=768,  or  i/-\-40x=z'7QZ 
From  (a)         ar=91--24y. 

Substitute  this  value  of  a:,  according  to  the  rule  in  equation 
^3)1  and  ^ 

y+40(91-.24y)=763, 
or,  y+3640-960y=:763; 

.-.  959y=:3640~763=2877, 
and  y=3. 

By  referring  to  equation  (a)  we  have  «= 91 -24y=f since 
'=3;  and  .-.24^=72)  91 -72=19.  !/     K     ^ 


Enunciate  the  second  method  of  solution. 


■mtm^m^x^ 


SIMPLE  EQUATIONS. 


Ex.8.  4:j;4.3y-_3i 


Sar+Sy 


=3!) 
=22  f 


6» 


Ans, 


=4 
5 


THIRD   METHOD. 

Multiply  the  first  equation  by  the  coefficient  of  x  in  tho 
Bccond  equation,  and  tfien  multiply  the  second  equation  by  the 
coofticient  of  x  m  the  first  equation;  subtract  the  second  of 
these  resulting  equations  from  the  Jlrst,  and  there  will  arise  an 
cquution  which  contains  only  y  and  known  quantities,  frhm 
which  the  value  of  y  can  be  determined 

It  must  bo  observed,  however,  that  if  the  terms,  which  in  the 
remlitnff  equations  are  the  same,  have  unlike  signs,  the  re- 
suiting  equations  must  be  added,  instead  of  being  subtracted, 
e'uatbns)        ""  ""^^  ^^  ^Uminaied  {i  e.,  expelled  from  tho 

Ex.  1.         Given  5z-f  4^=55  .  .  .  (l) 
3a:+2y=31  -  -  -  (2) 
To  find  the  values  of  a:  and  y 

Mult.  (1)  by  3,  then  15ar+12y=165 
"     (2)^7  5,     "  _15;»+10y=155 


.'.  by  subtraction,  we  have   2y=  10 

.*.  y=    5. 
Now  from  equation  (1)  we  have 

55~4y 


x=z 


5 
55-20 


Ex.2. 


_35 

~5 

=7. 


Let  the  proposed  equations  be 

ax-^byz=c    -  -  -  (1) 

a'x-\-b'x-c' (2) 

Mult.  (I)  by  a',  and  aa'x-{-a'by=za'e 
"     (2)  by  a,     "     aa'x-\-ah'y=ac' ', 


How  are  equations  solved  bj  the  third  method  f 


16 


in 


/5 


i-.f- 


■j^-mm": 


'SI  ' 


i 

I    ' 
I    I 


6  70 


ALGEBRA. 


0 


a'c-^ac' 


\ 

t 

c 


Mult.  (8)  by  4,  and  a'bx+hb'v=bc' 

B7  subtraction,       (ai'Z'b)lJtl_M; 

JtL  ■■       ab'-a'b' 

m'^'"'-*-  Let3.t+4y=29        (n 

>«!».  rn  by  3,  tl,e„  9?+l|z^?        (2) 
,   *l'-(2)by4,thenC8:.-lg:-fl4 

4-rfroi\^i:rtr,r:tr  -^  --'t'-  •••" 

together;  and  then  ^  equations  must  be  added 

77ar=231 ; 

rom  (])  ve  have  4y=29~3i?, 

=29--<),  (smce  ar=3 ;  and  .-.  3^=9) 
.•.y=:5.  ' 


ilx.  1 


'x.  5. 


^x.  6. 


=31) 

=22  f 


u^) 


!-x.  7. 


q  :.v,  8. 

n 


k.  9. 


Let    6.r+3y=:33) 
13^~4y=19f 

4ar+3y=:31 
3ar+2j/ 

3ic+2y=40  r 
2x+3y=35  f 

5a?~4y==iai 
4.r+2y=36  f 

ai?+7y=:79  ) 
2y--|a:=  9f 


^l?^+3 


r 


•    ■ 


If  ^ 


'J 


|y=4. 

\  y  ~7. 

(a;=ll 


/;  J- 


y 


)V 


-f  o 


5 


/i^ 


^!?. 

^i>.. 


li^ 


SnrPLE  EQUATIONS. 


Ex.  10. 


x-\-y 


3 

2x—^y 


23 


Ex.  11.  H^^+y=7 


^^7-^^^;=  ^ 


h'l^W^ti 


9 


5a;~-13y= 


67 


^r''/§./- 


-<4«s. 


Ex.  12.  2iziZy=2^+H-l 


3 


8 ~^=6 


5 


mn^.Lr^^f  /'""'^  unknown  quantities  are  concerned,  t'ne 
most  gor-oral  form  under  which  equations  of  this  kind  can  be 
expressed,  is  ax-\-by-{-cz^d        (I) 

a'a:+6V+c'«=c/'        (2) 
a"x+b"y\-c"z^d"       (3), 

and  the  solution  of  these  equations  may  be  conducted  as  in 
the  follow^ig  example : 

Ex.  l.#    Let  2a:4-3y4-4s=29     (1)   )    ,    ^  ^   ^        , 

3ar+2y+5^=32     (2)   '    to  fmd  the  values 
4a;4-3y+2^=25     (3) 


of  X,  y,  r. 


r.  Mi^^tiply /I)  by  3,  then  0.r+%+ 12^=87     (4) 
Multiply-X2)  by  2,  then  0.r+4y 4-10-^=04     (5). 

Subtract  (5)  from  (4)  then  5^+  2^=23     (a). 

Multiply  (2)  by  4,  then  12ar4-8y +20^=128 
Multiply  (3)  by  3,  then  12a;4-9//+  6^=75 

'ijj  Subtract     -    -    -    .     -y4-14z=53    (/3). 

^■^  JI.  Ilenrjo  the  given  oquations  are  rednnfirl  tn 

5//-f  2i'=23     (a) 


/5 


J  ^ 
<>   '' 


9 


L 


1ft 


7 


> 


,[**;"""  <» 


'^  5nip  ;»i->i,-/r 


^^^ 

^ 


ALGEBRA. 


72  • 

Mult  (S^f\  I   '   ^^+  2.=23 

By  addition    .    .    .     70^—900  ^         «, 

I'rom  equation  (/3)     .    .        „_,.      ^„      " 

\^)  y=  142—53=56-53=3. 

29~3.y~4s 


III.  From  equation  (1) 
Ex.2. 


29-2: 


Ex.3. 


ir+y+2=90 

2^+40=3y+20      ^ 
2ar+4O=40  4-lO      ) 

^-h  y+  0=  53 

ar+2y+32=105 
a:+3y  4-4^=134 


-o 


J*R0BLEM8. 

greater  kdd'ed  toldThe  t's  is7^u7l  T>  '^^  '  *'"^^«  '^- 
greater  be  suhtraL*  from  6  timl.  L  f^'  ^"^  ^^  ^^^'^^  ^^^ 
der  divided  by  8,  The  q^otien?  ^  ^V""!'  ^"*^  ^^^  ^e"»^'»- 
numbers  ?        -^     '  '"®  quotient  will  be  4.     What  are  the 

Let  ar= the  (/reaier  number 
y = the /e«5  number;    ' 


Then  3«+|=:36 


^jj.   9:c+   y=108 
'  6y-.2a;=  32; 


Or,  y  4-90.= 108     (1) 

W  then6y-  2a:=  32; 

then         56,r=616- 
610  ' 

56  — 

From  equation  (1)       j,=  108Jo.=,08-09.,9 
...  ru"""'  ^-  ''^eie  is  a  certain  fraction.  ,,..1.  ,i,„.  .-,•  ,  . .,  - 
".  "-  r.um.™to,-, ,«  ™i™  will  be  jd  ■  and  if'r;ubtmro,« 

4. 


-".='WMC(*BrtWMHi«*H', , , 


SIMPLE   EQUATIONS. 


73 


!^ 


from  the  denominator,  its  value  will  be  |th.     What  is  the 
traction  ? 

Let  a;=its  numerator,    )    ,       .,/.,.      .    x 
y=  denominator;    \  *^^"  ^^^  ^^^«*^«"  '^  - 

rr+3_l  ^ 

~3 


Add  3  to  the  numerator,  then 


y 


3^+0=  y 


^  .r        1 

Subtract  one  from  denom'.,  and  — ^ — =- 

y—l     5  J 

By  transposition,  v—^x—9     (1) 

y-5x^l     (2). 

Subtract  equation  (2)  from  (1),  and  we  have 

2x==8 ; 

.*.  2r=-=4,  the  numerator. 

From  equation  (1)  y=9+3j;=9+12=2l,  the  denominator. 

4 
Hence  the  fraction  required  is  — . 

21 

Prob.  3.  A  and  B  have  certain  sums  of  money  ;  says  A 
to  B,  Give  me  £15  of  your  money,  and  I  shall  have  5  times 
as  much  as  you  will  have  left ;  says  B  to  A,  Give  me  £5  of 
your  money,  and  I  shall  have  exactly  as  much  as  you  will 
have  left.     W  hat  sum  of  money  had  each  % 

Let  a;=A'a  money  )   ,      ^  •  i  k_  j  what  A  would  have  after 
y=B's  money  f  ^"^^  a;-|- lo-  ^     receiving  £15  from  B. 

y~  15= what  B  would  have  left. 

Again,  y+  5=  \  ^^^^  ^  would  have  aftei 
°     '  ^  (      reccivmg  £5  from  A. 

x—-  5=:what  A  would  have  left. 
ITencc,  by  the  question,  a;+15=5x  (2^— 15)=5y— 75, ) 

and  ?/-}-  5~iK-— 5.  j 

By  transposition,  5y—  a; =00       (1), ) 
andy—  a;~  — 10  (2).  f 

8ct  down  equation  (1)  5y—  x=90. 
Multiolv  eo".  ^2A  by  5,  5?/— !>i 


__  n{\ 


Subtract  (2)  from  (1)  4ar™H0; 


n 


'I 


74 


|i    ' 


ALGEBKA. 

•  .    ,7*"-":  0(r        1  f 

A  — 'J»,  A  s  money. 


4       •-"?  ^1.  s  money, 
■^rom  equation  en  ^//-ort  . 

.  ..     125 


J^KOB.  4.  Whaf  ^  ^       -•>.  Rsn.oney. 

,,„o  ?  '^'--"-  '  -btract  o«.,  ,h,  reUld:?^i;I„';7'' 

PROB   5    THaj-o  •  "^"*'-  ^»  '^''^J  3- 

to  its  mmevator  itV  '''''^'''",  ^^^^^^'^n,  such  th.,t  if  i    7 , 

■"^ggarsdid  he  relieve  ?      '"  '"  '"^  P^^ket;  and  howf,„V 

Tl-en  2ixy,  o,f  =  J  No  of .,;«,.  ,,1,,,  ,„„,^ 
"d  2Xy,  or  4=  '      '"'".^  S'*'™  »'  2»-  6i  each 
Hence,  by  the  q„ostio„,|'=,^^3     (1)  "' ^4,  each. 

Sub'.  (2)  fr„„  (,)  „,     y_  ^  > 

^??.?.  56,  and  33. 


It) 


Proc.  8.  Thoro   k  ^^''-  ^^»  ^ 

<^'git«.     Tho...,of  thnJS;"."'i'"^^«'S  consisting 


of    ?.«i.-rt 

^0  added  to 


t 


'****'««*'™i»»««**««««i,i 


^ngmmmt^-^sm,,;: 


SIMPLE  EQUATIOXS. 


75 


io: 


A. 


What  is  the 


St Zh^"""  '"""■'  *'  *8its  will  be  inverted. 

right.ii.d  diguTl's:  ^JiTxi'A!''^''""  '^«"  i^'"^ "- 

Let  xz=le/t.hand  digit, 
y-riffhi./iand  digit. 
?  ,^^-^+2'=the  number  itself! 

Hence  bvfht !."";'  ""^"^'^  ^^'*^  '^'g't^  ^>^''^^e<^- 
on.i  in    f      '^-^  *"®  question,  .r+y-5  nr 

Subtract  (2)  from  (1),  tl.,n  2y=G,  and  ;~|"-~'  ^^^^ 

.rzrS— y  =  5— 3—2  • 
Ajji^         ,•'•  t-^c  number  is  (10^+vW23  ' 

Add  9  to  this  number,  and    t  betomTs  30  whJ.^  • 
number  with  the  diffits  inverted        ''''''"'^^  *^^'  ^^'^^  is  '.ne 

^the, ..eater,  the  remaind^'^ajT^J^r;^^ 

the  digits  will  be  '■»..«     wLTtttLefuSr?""'''"'  "'• 

-t  be  ta^e  to  r.™  »  2^  ^'JZ.  SSttS 

^«5.  33  at  13.V.  (id. 

Prob.  12.  A  vessel  CO  imininr  ion      n       ^.^  ^'    ^■'''-  ^*''- 

mimitesbytv.ospol^rng^^^^^^^^^  '^   ^0 

gallons  in  a  minu  c,  the  other  O^lull'  '         ^"^  '''''''  ^'^ 

what  time  has  eac-A'spout  run ?      ^       "'  '"  *  "^'""^^-     -^'"• 

Ans.  14  gallon  spout  runs  0  minutes, 
Ptton.  13.  To  find  thr^^  •  ■    '         "^"i"^  f""^  ^  minute., 

I  the  sum  of  tiL .  If^^^r ';::?'is  \  ^'lU'^  &''  -^^' 

with  Jfh  fhn,liff!.....,„„    /"^  M  "  ®"«^1  ^«  120;  the  secoml 

I  thoi.unr;^7he\h;;;— --^^  -d 

^l«5.  50,  06,  75. 


i 


t 


76  ALGEBRA. 


CHAPTER   IV. 

ON  INVOLUTION  AND  EVOLUTION. 

ON  THE  INVOLUTION  OF  NUMBERS  AND  SIMPLE  ALGEBRAIC 

QUANTITIES. 

53.  Involution,  or  "  the  raising  of  a  quantity  to  a  given 
power,"  is  performed  by  the  continued  multiplication  of  that 
quantity  into  itself  till  the  nu|ttUer  of  factors  amounts  to  the 
number  of  units  in  the  index-%fthat  given  power.  Thus,  the 
square  oC  a=aXa—ri^ -,  the  cube  of  b=bxf>Xb=b^;  the 
fourth  poioer  of  2  =2x2x2x2-- 16;  the  Jifth  power  of  3 
=3  X  3  X  3  X  3  X  3p=243  ;  &c.,  &c. 

^  54.  Tlie  operation  is  performed  in  the  same  manner  for 
simple  algebraic  quantities,  except  that  in  this  case  it  must  be 
observed,  that  the  powers  of  negative  quantities  are  alter- 
nately +  and  —  ;  the  even  powers  being  positive,  and  the  odd 
powers  negative.  Thus  the  square  of  +2a  is  4-2a  x  +2a  or 
4-40";  the  square  of  —  2u.  is  —  2aX  —  2a  or  +4a^;  but  thp. 
cube  of  — 2a=— 2aX— 2aX— 2a=+4a*X  — 2a=-8a^ 

The  several  powers  of  -  |  And  the  several  powers  of——, 


are, 

a     a     a* 

a     a     a     a* 
cube  =^X^X^==^a-' 


2c' 


Sq>i. 


> 


=  -2cX 


|4l»lil>— -—  A7-A7A7 


»Si:c.=&;c. 


a 
'6 


a 
V 


2c 
b 


-+4c*' 
b 


Cub.         t)^^~''()v^     2c~     °-" 


4th 
power 


2c'"     2c' 

___b__     _J_ 

~     2c^     2c^ 

=+___,  &e.=.&c. 


b_ 
"2c 


*1 
8c»' 

b 


ON  THE  INVOLUTION  OF  COMPOUND  ALGEBRAIC  QUANTITIES. 

55.  The  powers    of  compound  algebraic  quantities  are 


mj^        TiTi 


Involution  performed  for  simMe  al)?8braic  quautitioa  ?— 55.  How  are  tht 
powofii  of  comp  mnd  (luivntitJ  m  raised  I 


GKBRAIC 

to  a  given 
tion  of  that 
3unt8  to  the 
Thus,  the 
<6=6';  the 
potoer  of  3 

manner  for 
}  it  must  be 
s  are  alter- 
and  the  odd 
iX  +2a  or 
a^;  but  tho. 

s  of—--, 

2c~     8c»' 
b  b 


2c 


%c 


UNTITIES. 

mtities   are 


iiui  iimiiner  la 
Howr  are  tht 


INVOLUTION.  ^ 

it^lK:  (Anr^^f  "?^3"  o'  '■-  I^"'«  <•-  expound 


E-v,  I.  What  is  the  square 
ofa-f-26? 
a +23 

.      a'+2a6 

^+2ai+4i» 
Square  =a^+4a,^q:4^5 


Ex.  2.  What  is  the  cube  of 

a^—x 
a^—x 

-  a.'x+x» 
Square  r=a*~2a«a;-fp;,. 

a^—x 

«'~-2a*^+^ 
--  «*ar+2aV— a^ 

Cube=a''~-s„-..j.:j:3;;3-^rr7i 


-Ex.  3. 

What  is  the  5th  power  of  a+5? 

a  -i~b 

«'+  afi 
+  ab  +5« 

«*+  2oi+i^=:Square 
g  +       b 

a»+  2a'5+""^ 

a«+3a^6+  3^F^b'=:Ci^- 
g  +       5 

a*+3««5+  3a'^^~^3 
__+_5!'^+  3««6''+    3a//Vi* 

«^+4aV/Tl?^4a53+^/=.4th  Power 


t±J^il:i2S!±iO«V.«+5a6^+i^=:5th  Po«. 


T8 


ALGEBRA. 


Ex.  4.  The  4"^  power  of  a+Sb  is  a*+12a«6+54a*A'^+108ai» 

Ex.  5.  Tlie  S!7Mare  of  3a;'4-2r+5  is  9x*+12a;'»+34x'+20* 
+25. 

Ex.  6.  The  cwJe  of  Sx-6  is  27a:«-135a:''+ 225a: -125. 

Ex.7.  ThecMieof  a;'-2jr+l  is  z'-Qx'+15z*-20x''+l5x* 
— Cx+1. 

Ex.  8.  The  square  of  a+i+c  is  a'+2ab+b^+2ac-{-2bc+c\ 


ON   THE    EVOLUTION    OF    ALGEBRAIC    QUANTITIES. 

56.  Evolution,  "  or  the  rule  for  extracting  the  root  of  any 
quantity,"  is  just  the  reverse  of  Involution  ;  and  to  perform 
the  operation,  we  niust  inquire  what  quantity  multiplied  into 
itself,  till  the  number  of  factors  amount  to  the  n\imber  of 
units  in  the  index  of  the  given  root,  will  generate  the  quantity 
whose  root  is  to  be  extracted.     Thus, 

(1.)    49=7  x7 ;  .-.  the  sq.  roo^ of  49  (or by  Def"  15,y'49)=7. 

{2.)—b^=—bX—bX—b\  .'.cube  root ot—b^{^:z:j;z)  =  —b. 

.  16a*__2a     2a     2a     2a        4   /16a*_2a 
^  ^^8r6'^~3^^36^36^3^;**'  V816"*~36' 

(4.)     32=2X2X2X2X2;.-.  ^32=2, 

(5.)      a^=za?Xa?Xd?',  ,'.^a^=a\ 

Hence  it  may  be  inferred,  that  any  root  of  a  simple  quan- 
tity can  be  extracted,  by  dividing  its  index,  if  possible,  by  the 
index  of  the  root, 

57.  If  the  quantity  under  the  radical  sign  does  not  admit 
of  resolution  into  the  number  of  factors  indicated  by  that 
sign,  or,  in  other  words,  if  it  be  not  a  complete  power,  then  its 
exact  root  cannot  be  extracted,  and  the  quantity  itself,  with 
the  ladical  sign  annexed,  is  called  a  Surd.  Thus  -y/37,  }/a^, 
V//,  J/47,  &;c.,  &c.,  are  Surd  quantities. 

.«.  Wha"  {mlMluMon?  How  is  it  performed  ?— 57.  Wlint  ie  a  Surd  qiumtity  f 


EVOLUTION. 


79 


derived  from  thos"  of  ZZlT  PoTin"  ""  T'  f  ■''^  "^^^ 
Ex.  3  ,  the  square  o{a+!ua'+<laf,In  T'''''  i^^  ^'-  ^S, 
arranged  according  10^0  nowtflf''"  '  7^"'^  ""'  '«'™s.-"-e 
with  a'+2a4+4',  we  observe  fh,.  ?K    «    .""  eomparing  „+/, 

tllf  Tr-f'''tsrtet';!?t-^'  '-">  "^"-^  ?-- 

root  (a).     Put  a  therefore  for  the  fir^f  «s_lo  a  .  w  / 
term  of  the  root,  square  it,  and    ubtrac    J+^"^+*'  (-+' 
that  square  from  the  first  term  of  the 

2^aT;.»      1"?  ^^J"  f^^  °'^^^  t^^«  terms  2^4T|2«i+i« 
-2«6+Z> ,  and  t/oM6/e  the  first  term  of  the  l2ab+b' 

root;  set  down  2a,  and  having  divided  ~^^ 

he  first  term  of  the  remainder  (2a6)  by  •==— • 

It,  It  gives  b,  the  other  term  of  the  root- 
and  smce  2ai4-A«— r9«a.;.\;.  -c  1    «     ', 

being  subtracted  fr&Vo^tL'"';l:ht'^'+'"  \*'"'' 
remains.  ™^  oi  ought  down,  nothing 

60.  Again,  the  square  of  a+b4-r  ( Av^  f^K  v     c.  x  . 
2ab-^b'4-2ac4-2bc4-r^-    in    7^-         (Art.  55,  Ex.  8.)  is  a"-+ 
deriTed  from  t    '      '    '^'''    "^^"    *^^    ^^ot    may    bo 

the  power,  by         '','+2«*+*'+2ac-f  25c-f  c'/a-f  b-i-c 
continuing  the    «  "■  7.^  \  ' 

process  in  the     *^"  +  X"*  "^  *' 
iast    Article.  goi+i' 

Thus,  baving  2a-^2bi-d2ac+2bc+c 

found  the  two  }ftac+2bc~  -'■ 

first  terms  (a+i) 
of  the  root  as 


befor 


4'i\t%ry^t 


f»_- 


ri ' 


'iOC  f-HOf 


»i?SVttSl«t»:;  jr* '''''""'^^^*^ 


'\ 


80 


ALGEBRA. 


i;,creased,  and  it  ^f ^X+st  +  J  ^hiXSg  subtrac.od 

iklhU  manner  the  following  Examples  are  solved. 

Ex.  1. 


4** 


4:c'4- 


|>'+T 


89  , 


6x»+|a:' 


\    20.tM-15a;+25 


Ex.  2. 

^^ . 

2a^+2a;')4a;'+2a;* 


Ex.  3.  Tne  sqaare 


rootof4x'+4a:.y+2/'is2a:+y. 


„.    .    rn,_.n„.«rnotof25a'  +  30a6+9/ris5a+36. 
^"  "  ^"^'  " "  ■" '    root  of  9.*+12.«+22.'+12.+a 


Ex.  5.  Eind  the  square 


^Ins.  3x'+2a;+3. 


__  *^^^^^^^      ■-■ai. 


•»--^ 


►.-wfci-j 


EVOLUTION. 


81 


»,  it  gives  f,  the 
term  (6)  of  the 
the  divisor  thus 
iltiply  this  n*vr 
being  subtrac.od 
res  no  remainder, 
solved. 


-|x+5. 


Ex.  6.  Extract  tlio  square  root  of  4a;*~-16x»4-2  ic'—Wx-^ 

Ans.  2x^—4x-T  2. 

Ex.  7.  Find  the  square  n  jt  of  SCir^— 3Gi-^+17^«--4jr4l 

y 

o 

Ex.  8.  Extract  the  square  root  of  «*-f  82;'+  24-f — -f-i?. 

a;-      «* 

4 

ylns.  ar'+4-j — . 


2a;*-a:-f2 


■4 
4 


is  2;c4-y- 
-9P  is  5a +35. 

Ins.  3x^4-2x+3.    . 


ON  THE  INVKSTIOATION  OP  THE  RULE  FOR  TJIE  EXTK  N 

OF  THE  SQUARE  BO^  T  OF  NUMBERS. 

Before  we  proceed  to  the  investigation  of  this  Rule,  it  wili 
be  no'cssary  to  explain  the  nature  of  the  common  arithmeti- 
cal  notation. 

01.  It  is  very  well  known  that  the  value  of  the  ficrures  in 
the  common  arithmetical  scale  increases  in  a  tenfold°propor. 
tion  from  the  right  to  the  left ;  a  number,  therefore,  may  be 
t  xpressed  by  the  addition  of  the  units,  tens,  hundreds,  &c ,  of 
which  It  consists.  Thus  the  number  4371  may  be  eipressed 
in  the  followmg  manner,  viz.,  4000+300+70+1,  or  bv  4^ 
1000+3x100+7x10+1 ;  hence,  if  the  digits*  of  a  nuU^r 
hanHhen  ^  "'    '  ''  ^'  ''  '^°''  ^^S^nning  from  the  left 

A  No.  of  2  figures  may  be  expressed  by  lOa+i 

«      3  figures  '  by     lOOa+lOJ+c. 

4  figures  «  by  1000c +1006+ lOc+cf. 

<^c.  &c.     &c. 

62.  Let  a  number  of  three  figures  (viz.,  lOOa+105+c)  be 

*  \^^^^/!!^'J^  of  ^  number  are  meant  the  figures  which  composa 
n,  cons,(?ered  independentlv  of  the  value  winch  they  possess  in  thHSh^ 
metical  scale     TJius  the  cTi^it.  of  the  number  537  aK  simply  the  num- 

Zn'nf'  ^1^^'  ^\^''^^'  **^"  ^'  ^on^i'lered  with  respect  tJits  place  n 
the  numeration  scale,  means  500,  and  the  3  means  30  ^ 


HiS;?  S  Gi  ^\?'"™?"."'''*""®*'°''^  s^"'^  of  notation.  What  ig  a 
digit  ?-b2  ShoNV  the  relation  between  the  algebraical  hfid  ml  nerioJ 
methoi  of  ox^raot.ng  tbo  square  root,  and  tliut  tfey  are  Jden  icJ 


/I: 
f  r 


m 

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82 


ALGEERA. 


squared  and  its  root  extracted  accordiiig  to  the  Rule  in  Art 
00,  and  the  operation  will  stand  thus ; 

I^00a'+2000a5  +  100i'+O00ac+o05e+cXl00a^ 

20(k+l0Z;)  2000ab  +  l00b' 
2000ab_j-im^ 

200a+206+c)  200acH-206c+c« 
200«c+20ic4-c2 


* 


*     * 


II.  Letff=2]      ,   , 

6=3  [  ^''?  ^'\^  operation  is  transformed  into  the  fol. 
c_lj      lowing  one; 

^^^^^+12000+900+400+00+1^200+30+1 

400+30)l2000+900+400 
/ 12000+ 900 

400+G0+l\400+60  +  l 
7400+00+ 1 


*     * 


5330i/231 


II.  But  It  IS  evident    that  this   operation  would  not  be 
affected  by  collecting  the  several   nJimbcrs  which  star d  m 
the   same   luie    into   one    sum,   and 
leaymg   out   the    ciphers   which   are 
to  be  subtracted  in  the  several  parts 
of  the  operation.     Lot  this  be  done  • 
and  let  two  figures  be  brought  down 
at  a   time,  after   the  square   of  the 
first  figure  in  the  root  has  been  sub- 
tracted; then  the  operation  may  be 
exhibited   in   the    manner    annexed- 
from  whwili  If  ni-»i->/->r.-..„  «^i,„i.  iL  _  ' 
;. trj-j."^«i3  tiiui,  uic  square 

loct  of  53,361  is  231. 


4m  urn 

|l29_ 

4011401 
1401 


] 


.  ^»— «?<#|«l|!l,'««!i-; 


'''^»«^<^>^^^^- 


QUADRATIC   EQUATIOXS. 


83 

square  root  of  100  is  10-  nf  lAonn  •  inn  '  r  '  ^'"^®  ^''^ 
1000  &c  &.ro  .  v.  f  11  '  ?,  ^T^^  '^  1^^'  of  1,000,000  is 
^""^t"  100  ;,  ,  t^^^^^^^  T'^<^  root  of'a  n'un.bor 

listing  of  rr%ures         '^^' ^^"^''^  ^«°'  ^^  ^  '--^er  ccn 
£x.  1.  Find  the  square  root  of  105,625.  .  Ans.  325. 

Ex.  2.  Find  the  square  root  of  173,050.  A71S.  41Q. 

Ex.  3.  Find  the  square  root  of  5,934,090.         Ans.  243G. 


CHAPTER    Y. 


ON  QUADRATIC  EQUATIONS. 

<U.  Quadratic  Equations  arc  divided  into  pure  and  adfec/Pd 
v/i^r,7^  of  the  unknown  quantity,  such  as  x'^SG-    ^4  5- 

«  ,U         i^'i  *^''  .  ^'^'^^^^'^^  ^"^^^r^tic  equations' a,  etlio'e 
UHch  involve  Ijoth  the  square  ind  .v.-.«7>/. ^o^m-  oHhc  un 


I  It 


i 

ill 


■^'.:>:?mf^mwrjmm..: 


84 


ALGEBKA. 


r 


ON   THE    SOLUTION    OF   PUUE    QUADRATIC    EQUATIONS 

necessary   by  the  coefficient  of;.';  then  extract' ttf    ^ 
root  of  each  side  of  the  equation,  and  ifwrnTvl  the  tC 

Ex.  1. 

Leta;'+5=54. 

By  transposition,  a;^=54— 5=49. 

-Extract  the  square  root ) 

of  both  sides  of  the  f  then  xz=  +  W49--^7 
equation,  j  —V^^-—'. 

Ex.  2. 
<         L'et  3j;'— 4=71. 
By  transposition.  3;c'=71 4.4=^75^ 

Divideby3,a:''=^=25. 

o 

Extract  the  square  root,  x=j^^25=+5, 

Ex.  3.  "~ 

Let  ax^—bz=ci 
then  ax^=zc-{-bf 

and  :r*=f±i 
a 

V     a 
=244  -    -    Ans.  x=  +7, 
=3a;'+63     Ans.  x=z-{-^^ 

Ans.  ar= +  10. 


Ex.  4. 
Ex.  5. 


Oar' 4. 9 
4a;'+5 


9 


=45 


Ex.  7.        i;j 


65.  State  tha  rn\a  ^nr.  o«I..: .  ~~ — 

._  .-..  ^..-iTiuj,'  juiro  qxiauruiio  oquatloug. 


of  the 


QUADRATIC  EQUATIONS.  flg 

ON   THE    SOUTIOK   OF   ADPHCrED    QUADRATIC   B0UAT.O.,-S 

y       or  /  actional.     Divide  each  side  of  this  equation  l.y 
a:  + -a;  =-.      Let  --=;9,  -  =q  ;  then  this  equation  is 


KULE  I 

Let  x^^px~q. 
Add  |!  to  each  -<3e  )    ,^     ^^^^^.  ^,^^^ 

of  the  equation,  then  j  4      4"^'^  4 

Extract  the  square  root 
of  each  side  of  the 
equation,  then 


andar=il:5!+47+7> 
2 


will  arise,  on  the  lejjnd  sld/'of  tt  "Sn'  f  ^'  "^•''•^* 
which  IS  a  complete  square;  and  by  pvI^?;"!  Z^'  ''  '^"•'"'^'^>' 


binco   the  square  of  +«   fg   ..„?   „„j     », 
of  P=+4,  ™ay  be  expressed  by  ±  /,?:^  '  ''""  ""  ^^"-^^^  '-* 


reM^irr^fry:;- Pjj'^JlJ^n^  Can  U  b. 

8 


86 


./, 


!i 


-ALGEBRA. 


one  corresponding  to  .he  sic^n  1       i  f, ^^ill  have  two  values ; 
of  the  radical  quan%        ^"  "^'  '"^  '^^  ^^^^^^  *«  the  sign-,' 

Ex.  1. 

.  . ,   ,  Let  x'+8x=Gr\ 

^^  Add  ft.  y^<'Jl^J^^O)^a^  ,Me  of  the  equation, 

E.U.-aeUHos,„a;etor„JtKrrl  e,u.io.  .1 

ar-f4=+ V'81  =  +a 
and  a;  =  9  —  4  — "s".  ' 

or  a;  =—9-4- -13. 

Ex.  2. 

Let  a;'-4ar=:45. 
Add  the  square  iof  ) 

2  (e.  e.  4),  then  f  ^  -4a;+4=45+4=49. 

Extract  the  square  root,  and  x-2=  +  y'49=  +7 

anda;=7+2=9;  ~~  * 

or,  ar=2-7=:-'5 

-  ■    ^«*-  ^=  6  or -18. 
X — 14e —  '1I                           J 

.  —  01   .    .    .  ,     ji^g^  ar=17or—  q 

a:'-  Gx=  40  .                     ^  '  or        J. 

^^  -    -     -  -     Ans.  a;--=10or  —  4. 

Ex.   0.  a;« 5a;=::(J 

In  .his  exa„,plo  ,ho  oo.^ri«,  „f  ,  ;,  5,  „„  ^,^  _^^^^^^^^ 
ts  hains  -;  a„d  ...  adding  to  each  .ide  of  the  eq„a«„„ 

(2)  ""  T' "'"  e«t 

I 

x'-~-5x+(^Y=G4-^-^+^^    40 
\2/        ^  4  -~4      -^  J. 

5      _l7 


Ex.  3. 
Ex.  4. 
Ex.  5. 


extracting  the  sjuar*  root,  ar-5  -  +7  • 

2  "~ — o  ' 


2     ~2 

_54.7_^ 


2-2 


r— O,  or  —I. 


■pwnciisiaWkiS 


■■.,  .«?-i^i^S^«ft^^ 


QUADRATIC  EQtJATIOXS.  3) 

Extracting  the  square  root,  x~l=-h^ . 

^     — 2' 

1-4-5 

•  •  ^=^±2=3  or-S. 

«r-^-/ar_7» Ans.  xz:zQ  or  ^IS. 

Ex.    9.        a;24-3^=28    .    .  j  . 

Ex.  11.         a;2j-  aj—on  , 

Ex.12.  Let7^«-.20^z=32;  find  or. 

Dividing    by    7,  x^-^^x=^-^. 

^:pJ;;  \  .^-^.+  (^y.L!52.^^4     100,324 
I  J  7        V7/       7^41)       40^49 -4ir* 

Hence,  :.-:L?=  +  ,/??4_  +  18, 

anda:=— ±l?=4or-i|. 

Ex.  13.     •        6x«+4ar=273. 

Dividing  hy  5,  a;«+-^=?Z2. 

5        5 

.idfadd' ^  I    (%v±  and  .'+ t+l-?I?  .   4  __  1369 
^     W      25  5  ^25-  5  ^25-~25^ 

"•      Extracting  the  square  root,  «+?-  +?Z 

'    ^5~— 5* 

Ex.  14.        3a''+2^— ini  A 


ii:. 


S8 


ALGEBRA. 

Ex. 
Ex. 

15. 
10. 

-5z= 
-2x= 

=117        .     .     . 

=280      .    .    . 

Am. 
Ans. 

■•  Oor 
:10or 

-  0* 

Ex. 

17 

4a;'- 

-7x= 

:49?       ,     .     . 

Ans. 

X=z 

■12,or 

3 

-101, 

A  qt.adratic  equation  seldom  aonpnr^  in  o  ^  .      . 

as  those  of  the  preceding  exaTples^  ft  I  t)f  7""  '^  ''""^'^ 
foundnecessaiTtoemnrorinT  'i  V  ^^^^^^^^'^  generallv 
following  reductiol.^    ^      '^''  '°^"^^^"  ^^  ^  ^^^'^^''^tic  the 

(1.)  Clear  the  equation  of  fractions. 

(2.)  Transpose  the  terms  involving  ^«  nnrl   ^  f^  .u    ,  .v 

ha«d  and  th„  numbers  to  the  ri«hth^„/.4"V.he%;t:^ 
^^,V3.)  B:v,do  all  the  tornas  of  the  equation  by  the  ..V.C 

(4.)  Complete  the  square. 


Ex.  1. 


4a;« 


1 


3         ll=r- 


Multiply  by  3,  and  4a;'-33=.r. 
By  transposition,       4.r—  a; =33 

Divide    by  4,   and    x'-.\x~^ 

4       4' 
Complete  the  )  1        1      33      1      ^oa      , 

square,         \  ^"~7^+;rr=^+i-=2l24.1_ 

^  4      04     4^64     64^64- 64 

Extracting  the  sq.  root,  x—-=:  +?? 

■.8—8" 


529 


I4-23 


Ey.  2. 


9       4 


9+- 


4rJ-J. 


a; 


•=5ar-f-5. 


•«*«^:«i»«SR%g|^:-?^, 


Ex.  3. 

Ex.  4. 

Ex.  5. 

Ex.  G. 
Ex.  7. 
Ex.  8. 

Ex.9. 

Ex.  10. 

Ex.  11. 
Ex.  12. 


both 


—4 


of 


QUADRATIC   EQUATIONS.  gft 

5      25     5^2o~25' 

5     —5' 

andar=-+-=2or-l 

6      ^--*+lI Ans.    x=:zl2or-6. 

2^     1     7 

3  ~*"^'^3*  •    -    •         -    -    Ans.     x=3  or  J. 

3      2~" ^ns.     x~G  or  —1. 

2 

a;+l'^a:"~ -^w*«     a?=2  or  — ^. 

a:»-34=|^. Ans.     x =6  or  -^l 

5+- =5] Ans.    ar=25  or  1. 

,    24      ^ 

'^'^^^ZiT^^''''^-      -     -     -    ^«^.     a:=5or-2. 

a;+l        a?    ~"6"      '    *    ■    ^"*'     ^=2  or -3. 

x-\-2       0    — ^"-^•-    -    -    Ans.    ar=10or— i 

Given  a;'4-8a:= -31;  find  ar. 

^+8a:4-lG=lG-31  =  ~15, 

x-\-4=  +  V—l6; 

/  — 15,  &  ar=  —4  —  V  — 15^ 

lich  r-re  impassible  or  imaginary  values  of  x, 
-  8* 


90 


ALGEBRA. 


Ans.     ar=1+^V  — 1^ 
Ans.    x=l5  or  1. 


Ex  13.       a;'-  2:,=  -  2.      .    . 
Ex.  14.        £'-lQ£=z-lo.        .     - 

Ex.  15.  Let  132«+2t=G0. 

Divide  by  13,  3;''  +  ?:?=^. 
^      '     ^13     13 

square  of  ^  j  -  +T3^1oi)=l3+nu>=I(i9+loi)=10 


Extract  the  ) 


781 
109 


1 


.4--V/T81 


square  root  r^"^i3~ —    73" 


=,+ 


•    •    aC    ■  I- 


±27.94-1     20.94 


27.94 
•   13 


13  ^  Is"^^-^^  ^^  -2.226. 
Ex.  16.  ar'-6.r+19=rl3.  -  Ans.  a:=4.732  or  1.268. 
Ex.17.     5.c=-f4.r=25.         .    .     Ans.     ;r=1.871. 

_  Any  equation,  in  which  the  unlcnown  quantity  is  found  onlv 
ni  two  terms,  with  the  index  of  the  higher  power  double 
that  of  the  lower,  may  be  solved  as  a  quadratic  by  the  pre 
cedmg  rules.  J         i 

Ex.  18.  Let  a;'— 2^^ =48. 

Complete  the  square,  .r*— 2.c'4- 1  =49  • 
Extract  the  square  root,    .r^— 1  =  +  7- 

.-.  x^=8  or  —6 ; 
and  .-.  X  =2  or yZTT 
Fi.  19.  2x~7^x=m.  ^ 

7   ,      99 
^-2V-=2-' 
•7    .    .  /7\«    99    49_841 
2  "^lO^Te* 


2»     ^U/      '>-      i«~^^ 


4     4  2 

•.  by  squaring  both  sides,  a- =81  or  ~. 


:-^mLmm 


QUADRATIC  EQUATIONS. 


9). 


; 781^ 

>r  -2,226. 
or  1.2G8. 


'oiind  only 
er  double 
y  the  pre 


Ex.  20. 
Ex.  21. 


^+4,r'^=12. 


-     Aus.     z=±^2  or  ±y/Z:?, 
•    Ans.     ar=3or^i:TJr 
Rule  IJ. 

Multiply  each  side  of  )    .       ^^^  ^^'±^^-^=<^,     ' 
tlie  equation  by  4a,    \  *"^"  'i«'-i'l:4a4.r=4ac. 
Add  b-  to  each  side,  )  .  ,     , 

we  have  f  4a  V+^  4adz-i  b^=4ac-i-l,\ 

Extract  the  square  root  as  before,  2ax±6  =  ± ^4^;^^ 

and  X  =i  V^^^TZ"'^:^ 

rom  which  we  infor   thni-  wp       ,     . 
U  multiplied  by  /„",•;«  1    "^  T^  ''^^  °^  "'O  "q^t^n  . 
side  there  be  Idl{  rJl^^T'  "^f:  ""<>  *°  '»'*  ' 
quantity  on  the  left.h„„d  sWe  of  1        ""■^"""  "^  "'  "'« 
square  of  2a.+4.     Ex  net  tt  f         ''™"°;'  "'"  ^^  ""-' 
tho  equation,  a"Sd  there  Irifes!  PI  ''''"'  "'^  *^''«'''  ^'''^  °<" 
the  value  of'.  „ay  be  determLd  "^'^  ''"'""'"•  '■™"'  "'''»'> ' 

i..'tL:7a!;*heSre°?he'p1"'="*  '".^^  '■°™  -•+P-^=?; 
Pl>ing  each  sTde  ofThe  equatt  Z\  ''' ^^'^^''^  ^T^-nuk 
of  the  coefficient  of  "  "  '  ^^  *'  ""•*  '"'''"'g  ">e  square 

spondingto  theSL  +  Id'lV^l  "^  ™'"'^;  <"">  corre- 
radical  luantityT      '         ""^  ""'"•  '»  *«  ^'g»  -.  of  the  • 

Ex.  I. 

Multiply  each  side  of  the  ) 
equation  by  (4„)  ,3 ;  j  36.-'+  60^=504. 

«ISt£S;I°  ,"!!:'*,'?.„«•■> -M.bo  found  In  .!,„»«,   ^„„.   . 


I 


^H2 


ALGEBRA. 


-^^cM  (/>«)  25  to  each  side  ) 

of  the  equation,  we  have  f  ^Cx'-f-GOjr  +  o-.^g^j^  ^25=520 

.-.  6jr= +23-5=18  or --28• 


28 
(J 


or  a-=-r-= _42 

Ex.2. 


14 
3 


Leta;'— 15jr=— 54. 
Multiply  by  4,  then  4^^-C0^-_oig 
■  Add  (i«)  225  )       ,,, 
to  each  side    [  ^"^  4x»-60x+225=225~216=9. 

Extract  the  square  root,  2ar— 15=+ ^9—4-3 . 

18      12  •'•2^=lS±3=18orl2. 

and;r=_or~==9orG. 


\ 


ON  inE  SOLUTION  OF  PROBLEMS  PRODUCING  QUADRATIC 

EQUATIONS. 

values  of  the  unknown  l;„t  ™'r  ""^  "f  ""^ 

'  lonniVa,!      m, .    .  "™'  quMtity,  Will  answcr  the  conditions 

.  3r.et:^l:.\7rnr:  :?;t  -h,t  i^i>r  - 

Problem  1. 

:  ,.-Ltsi;lVb:  cV '"  '"*° '™  ^-•'  p''^'.  "■»'  'h^-- 

Let  x—one  part, 
then  56— a? = the  o//<^r  part, 
and  X  (56-a:)=  ;;rocf«c;  of  the  t%Yo  parts. 
Hence,  by  the  question,  x  (56-.r)=640, 

or  56x— a:':=640. 


gUADRATIC  EQUATION-: 


H  f  25  =  520. 

equation,  whick 

i:23; 

8  or  —28; 


-n- 


1=9. 

[13; 
8  or  12, 


ADRATIC 

ve  quadratfc 
■  one  of  the 
e  conditions 
Mys  be  verv 
L  itself. 


s,  that  theii 


3. 


9S^ 


,j  ,    .        %  transposition,  a;»-5Cj:=:  -040 

ii>-  completing  the  square,  )    „        . 

(UuleI.)  j-^-^^-^-f784=784-.040z=M4 

anda:=^8  +  12=40or  10. 
Ill  th,^  case  It  appears  that  the  two  values  of  the  unknown 

I   If".?"-  ~'  ^"^  "^  '"■"  ""mbcrs  whoso  dilTerenoo  is  7  ai.fl 
nun^oei.     VV hat  are  the  numbers? 

Let  a;=the  less  number, 
then  «+7=the  r/reaier  number, 
and  - +30=3 half  their  product  plus  30. 

Hence,  by  the  question,  ^±Z)+3o=^.  (,j,,,,  ^f  ^^^^^^ 


or 


+30=ar». 


Multiply  by  2  .     z'-{.7x--].Q0  =  2x\ 

nf  u?!'  transposition      .  x"--7x=Q0, 

Multiply  by  4,  and  add  )  ,  ,,     ^„ 

49  (Rule  11.)  f  4^-28^;+ 49=240+49=289, 

/.  2a;-7=y'289=17 

2^=17+7=24,  or  ar=12  less  numL3r; 
hence  a:+7=12+7=19  preaier  number. 

Prob.  3.  To  divide  the  number  30  into  two  such  parts. 

Let  a'=the  /ess  part, 
then  30--ar=the  greater  part, 
and  30-ar-a;  or  30-2a:=their  c/^/e,-e«ce. 

1  Tf^nna    ^\\T  +Vi 


.  V  >  /r>/\ 


AV'^v— a;j=«x(oU— 2.ir), 
or30x-a;°=240-lCj:. 


I 

'i 


m 


I  i 


ALGEBRA. 


Bj  transposition,  a;«-w46^-- _24Q^ 

'bmplete  the  square.)    ,     ,^ 

(UvLEl)  [^-46^+529=529-240=280; 

.-.  ar-23=  +  ^289=  +  17, 
and  ar=23±17=40  or  6=/m  part  , 
30-ar=30-  6=24 =<7rea/*r  part, 
(n  thi8  case,  the  solution  of  the  equation  ijives  40  and 

.<0,  we  take  6  for  the  less  part,  which  gives  24  for  the  ZlaZ 
i^da/red  '  '''  '"^  """^'^'*^'  ^  ^"^  «'  ~  ^h^  tSS: 

sofen'at&'T  n''"^^^  '^^'l^^'  ^^'^  ^^^^  ^h'^h  he 
.oi   V  i'^  85.,  per  piece,  and  ga  ned  by  the  bar<raln 

as  much  as  one  piece  cost  hin,.     Req'uired  thi  number^of 

LetW=the  number  of  pieces. 
Ihen  — =the  number  of  shillings  each  piece  cost, 

.  ^^  f  r'l"  """'^''  ^^  '^^"^^^'^^^  ^^  *^^^^  the  whole  for; 
. .  4Sa:-675=what  he  gained  by  the  bargain. 


■675=5Z5. 

X 


Ilonce,  by  the  problem,  48j? 

By  transposition  )         225      225 
and  division,    J  *  ""'le'^^'io'* 


=  -7^+ 


1024  ~~  16  •   1^ 

.  ^_225_      /( 

32  ~V    1024 
255+225 


1024* 


225         /05025    255 
32* 

=15. 


and  X: 


32 


Prob.  5.  A  and  B  set  off  at  the  sam^  time  to  a  place  at 

^JT^^'l  '^  ?^^  "^"^^'  ^  t^^^^«  3  miles  anC>Z  faster 
than  B,  and  arrives  at  his  journey's  end  8  hours  aiid  or 
minutes  be/ore  him.    At  what  rate  did  o Jh  nToL"";!,!^ 

per  iiuur '^  ~      *         '"    •''"■=-'■ 


.>vmm.r^i«m: 


240=289; 

!89=i:l7, 
G=less  part  , 
realer  part. 

fives  40  and 
''  be  a  pari  of 
for  the  (/reader 
the  conditions 


15s.,  which  he 
)y  the  bargain 
le  number  of 


Be  cosif 

the  whole  for ; 


QUADRATIC  EQUATIONS. 


95 


5_65025 
'""1024* 
255 
32' 

15. 

0  a  place  at 

1  hour  fasier 
)urs  and  2C 


Then  'fUT""  r.  ^"'"^  "'„*"-''  B  <  -vels. 

And  — =number  of  hours  for  whi^h  B  travels. 
150 
x-\~S~~  "  **  "    A      « 

ney\" e^d't^L'er  "'  "'"'"  ^''  '""^^  '''"'''  ''  ^'^  J^^ 

i50  150 

150     25     150 

By  reduction,  x^+Sx=54. 

Complete  the  square,  ;i:«-f  3^-1- -=544.^. _ 225 

,      .  ox-t  ^__o4i-- ----(Rule  I.): 


Hence  -^l^Clj_fii_  ^^" 


or 


2    V     4       2  ' 


4       -  ■  4 

225     15 
f 


,        15--3 
and  x^ — —  =G  miles  an  hour  f)r  B, 


2 

a:-|-3=9  «  ^ 

flight'?h;L^e^'oo^tTfh^.t?  f.f '^  "P^^  *  *^^^5  at 'one 
Iths  of  ther  two  bees  ttn  5  *^^ ^^"i^^vay ;  at  another 
on  the  tree  ?*  "  remamed.    Honv  many  alightc^d 

Let  24;»=the  No.  of  bees 

then  xi-  ~-+2=2x\ 

or9ar+16.r»+18=18.c» 

.-.  18u:'-.16a;«— 9x=18, 

/r.  or  2.^''— 9a;=18 

(UuLE  II.)  Multiply  by  8 

16i:^--72.r=144. 

^^^./S'lb  Jl!^^-  .'^•-  M^  Strachey's  tran«lation  of  t,.. 

pudson.  be  foumi^t.;acr:mrwmw£%/;-'7'"  7^7"?"  r"''  "P*'"  ««'°' 
8ta..d.  h,  that  tnu.slution.  p  02  '^''"'^  °^  '^^"^•°"'  «»  i< 


96 


i  I 


<■? 


II 


I  !    i 


ALGEBRA. 


Add  81;    then  lG^»-72ar+81=225, 

or4x~9=15;' 
.•.4^=15+9  =24, 

*  „  ^        24 

and  ir= — —a. 

4        ' 

•••  2^'=72,  No.  of  bees. 

tnlTrolIt''^^^^^^^  ^6^ "'"  ''  "^^^  ^^^.  -^^  P-*^  '^-^ 
P         o    WL  ^iw«.  27and6. 

JlZLY^I '""  """"'""  «^^  ""'-  -h"^  -m  is  29, 
p         Q   rn.  ^''*-  25  and  4. 

Of  tS  AS  f '^rwtTo  rn\^x^.V"^  *'"  ^-^ 

p         .-    _,  ^««.  13  and  8. 

.?xa„  „/«/«  ^.,„,,,.    Vhat  fro  th:"'u™beri  ?'  "^'""  '°  "'" 
P         11    m.  ^«s.  17  and  11. 

p         ,„     .  ^  ^««.  15  and  8. 

B  gives  £5  aV„e  to  each '^I  ^'IP^hT;"  H  "  '''  "'"^ 
persons  were  relieved  by  A  and  B  rj^ecttve^-j        "  """-^ 

^«».  120  by  A,  80  by  B. 

We  cost  him  TshmLsTss      H^'  '""^  '^^^  """^ 
there?  sumings  Jess.     How  many  sheep  weru 

for^S'7.-    HavinStsTs  o?"f^  "  ^r'"  "'"»''^'  "^  ^''eep 
8  shillings  a.ird%'„fit\:^-:''.rC- ."''.''•f  ""■"'""'^  «' 


many  sheep  did  he  buy? 


igjiin.     How 
Am.  38. 


4 


'  bees. 

0  such  parts  thai 
Ans.  27  and  6. 

hose  sura  is  20, 
4aw.  25  and  4. 

s  5,  and  jth  part 

ers? 

1/is.  13  and  8. 

is  6 ;  and  if  47 
be  equal  to  the 

i? 

!«.  17andll. 

sum  is  30;  and 
uare  of  the  ksj- 
ns.  21  and  9. 

product  is  120. 
>m  the  greater, 

1  also  be  120. 
''«.  15  and  8. 

mong  a  certain 
>re  than  B,  and 
-     How  many 

A,  80  by  B. 

ibcr  cf  sheep 
sheep  u-ould 
Y  sheep   were 
Ans.  40. 

iber  of  sheep 
remainder:  at 
argttin.     How 
Am.  38. 


QUADRATIC  EQUATIONS.  97 

rhe'^di^tlcLfso^^  to  a  place  at 

an  hour  faster  than  T)  .  d  i  vesttl'    '  ''''  ''  ""  ^"^" 
hours  before  him.     A'"  ,'\^^fllZ%A^  ^u  J^"^"^^ «  end   10 

Let  ar=6ji^  part, 

t^enl6-rc=t't^' 'other  part. 
Hence  x  (16-a:)  or  16x-^^?rt70. 

Transpose,  and  ar'—l'diii— 70. 
Complete  the  square,    '  *.'.'.' 

a:«~  16^+64=  ~r0.4^iij4==  _6^ 

.-.  a;--8=:  +  -v/irO,  or  a:=8+>/IlG» 

Let  ar= one  factor;f  .'* 
then  -=the  other  factor. 


Hence  x'^+^—h 
x"      * 

or  ar'^-f  a'^ar*. 

fro;|tV;Sp;SttTt?e^*J;;^^^^^^^  -'-^^  can  „n. 

may  be  divided  is  when  these  two  lZ\l  iT  'f  ""''J  ^'^'^^  ""'"l^^*' 
jluct  therefore/winch  couIdTrir?romn,rr''?""^'  I''"  ^'•^»*^''t  P'"» 
•nto  two  part.,  is  when  each  of  Is  8 'Tn/^- *'''  ""•"?'"'•  '' 
(livide  the  number  16  into  two  si  c ,  Trt-  ♦!  *  *i  "''''•  '"  •"^q"'^"^  "to 
70  '  the  solution  of  the  querontr/r*):'  '''''  P^^^^^  should  be 

Plitd  tL<tw!S'!!:!_"!^""A  *»>«  two  numbers  which  bdn.  n,.,,,:. 


I 


!  ;  9 1 

■     II 


II 


98 


ALGEBRA. 


By  KULE  17. 


2  ^ 

^....  -^^5.  6  and  3. 

0 

ON    T,«    acuTIO^   OP    QnADRAT,0>SQCAT.O»S  COKTAi™  ,we 

UNKNOWN    QUANTITIES, 

ver/  well  kiown  these  cli.es,  the  two  following  are 

*  V 

Case  t,' 

equation;"  i  ^h  ch  case  X  P„^- ''t'!™'"'''''  '^  ^  "''"^'^ 
then  submitute  Lit  Z  v„i  ")■  ""*.'  .'™P'''  '^1»^'»».  "d 
-Mvcd  fey  th^e  :?S;  rI;;?  ^  ■J^^^^^'--  -W^h  may  be 

*)lwtion  of  these  equatiotcrntnf^t/'^^^  ^''^    ^^"^'•al 

of  higher  dimensio2rtharqt!;,Vralfcs  '*'^  ^^  '"'^'^"^  °^  '^'^''^'» 

equatio,..  to  one  quadr^i^^f 'ti;^  IZ^ZmT'^'"'^  ^'  '"^"^"'^  ^'«'  *^ 


)     i 


"^^■^^^'^^^mmk.:'^^0mm.^ 


o  two  such  fee 
^ns.  6  and  3. 

CONTAINING  TWC 


n  quantities,  in 
in  a  quadratic 
I  by  means  of 
)  following  are 


the  values  of 
3,  is  a  simple 
a  value  of  one 

equation,  and 
ther  equation ; 
^hich  may  be 


equations  cm 
his: 

t    the    general 
na  of  eqnationa 


Jclucing  the  two 


QUADRATIC  EQUATIONS. 
T    *       .  «  ^X.  1. 

Let  a:+2y=7  ) 

and  x^-\.Zxy-y  =23  [  ^^  ^"^  ^^^^  values  of  ^  and  y. 
From  1st  equation,  .=7~2y,  ...  a:«=49-28y+4r ; 
Substitute  these  values  for  x  and  x^  i„  the  2d  emmtion 
then  49~28z,+V+21y-.63,«-y=2S,  ^  ' 

R     I?         ir  °r  3/-+7y=49~23=2G. 

By  Rule  II.     3G/+a%+49=312+49=361 

:"./0y+7=19 

6y==19-7=12,  ory-2 
ar=7-2y=7--4=:3. 


99 


J  *  •  1  • 


Let|±-^=     9 


Ki,  2. 

to  find  the  values  of  x  and  y. 


and  3a'y=210 

From  1st  equation,  2a;+y=27  • 

.-.  2^=27— y. 

apd  xJitry 

'      '        2 
Hence,  3^y=3x^^Xy=210, 

or3x(27-y)xy=420 
8ly-3y«=420 
27y-  y»=140; 

R     T>         ,T  ory'~27y=_l40. 

By  Rule  H.,  4y«--108y+729=729~500=109  ; 

.•.2y-27=13,ory.?Z|:!3_.^3,>^ 
andx^^Lz^^^^ 


ill! 


!   If! 


j  1 1 


III  I' 

!     i      ! 


l\\ 


I  !' 


nil  I 


100 


ALGEBRA. 


.to'nuXl  '°  ""  """"  °'  "-^  '"^•'-•^  ^!g!'-    What  I. 

Hence,  a:=3y )  , 
and  lOjr+y— 12=  a;''  |  ">^  "^^  question ; 
.••  by  sub.)  „.     .  .'     . 

stitution   r^^+y-12=:9/,ffi,i.Mo^=,30y,and  ;r»=0/) 


0        3        0 

31         -12 

J: 


.-.  2,'--y= 


ByRuLE  I.,y'-?-v4-— -2^'    ^^    061-432    529 
9^    324~3^^^=-3oT-=324- 

,      Hence,  y-?^=??.  or  v--^'*~q 
«  '•'is     18'?^  ^-18=^' 

a^=3y=9^; 
and  consequently  the  number  is  9^.. 

E<4.       Let2ar-3y=   1)    *'*•' 

2a;*4-ary-5/=20  j  ^^-^"^  ^^e  values  of  x  and  y. 

•°'  ^ws.  ar=5,  y=3. 

35;  and  if  four  times  theelJ^rhl'^--/J"^""'r  *"'  l^» 
tl.o  less  plu,  one,  the  quotifr^n  vf  w  \  ^^''^  "'mo^ 

ber.     w'hat  are  'the  nSo™ /'"  "'  '''™'i'l  *?,'^"  """" 

-d«5.  13  and  4. 

^wj?.  78. 
Case  II 

71    When  x\  y\  or  a?y,  is  found  in  every  term  nf  fJ,.  . 
equations,  they  assume  the  form  of  ^  ^^^  ^'^'^ 

n.ay  ho  effected  :-as  in  the  following  E^ampiesr"  ^"^"""" 


^'^^^^fS^**''^' 


digit.     What  ia 

^rt.  61,  lOx+y 
liber. 

question ; 
rj,  and  x^—^xf) 


-432529 
24      -324' 

=3, 


es  of  X  and  y, 
ar=5,  y=3. 

if  the  less  be 
ainder  will  be 
by  three  times 

the  less  num- 
B,  13  and  4. 

i^hose  digits  is 

digits  will  be 

Ans.  78. 


w  of  the  two 


fk 


QUADRATIC  EQUATIOXS. 

Ex.   1. 

Let2a''-f3ary+y''~20 
5^+4^'=41  ; 

AMumex=.y,  then2.y+3y/+y^==oo  or ,/»- ?? 


101 


and  5i;V+4/=41,  or  v'-  -i] 


2i;-'+3y-f  i 
41 


20 


41 


Hence 

2y*+3?;+l  ~~^^p^^ 

which  reduced  is,  Qv'-^^\v^~.\i. 

41v__^I3 
"  6  ~     "6"* 


5z''''+4' 


.•.i>' 


%  KuLE  I.,  z/'-iJf  .lggl_13C0 
6        144       144  > 

.•.t.-i^=:±2!.or.-41±37     13      . 
12       lo>or._-__^^_,,.. 

Let  v=5,  then  v«=-^L__  ^1  _369 

v=2;y=^X3=l. 
,  Ex.  2. 

theg.tlH:77Ta'„TXse'dT^  ^"^  -"^^'P^-d  by 

i«  equal  to  12  ?    '         ^""'^  ^'^^^^«^«  multiplied  b^'  the  less 

Let  ar=greater  number.      ' 
2/= less 
Then  (^•+y)X:r=;^r»4:a-v=.77 

Assume  x-=.vy' 

Then  j;y+^y.^77  )  or  /-^  j 
andt,y«-y^=12'[  if" 

<ii 1 


w 


M 


M  ! 


i       I 


1      *l  ' 


ihl 


102 


ALGEBRA. 

or  12v«4-12v=77i;— 77; 

65  77 

which  gives  v* v=i--  — 

19.  1 0 


12 


12  ^576~57G' 

.  „_65±23    88  or  42     11      7 

24  24      ~  3    "^4* 

£'////«•  value  of  v  will  answer  the  conditions  of  the  question , 

'''     -  12        12        48      48 


but  take  v=-.;  then  i/ 


v-l     1-1     7-4~~3""     ' 
and  y=4, 

7 

a;=vy=:-x4=7. 

4 


Hence,  the  numbers  are  4  and  7. 

i 

Ex.  3.  Find  two  numbers,  such,  that  the  square  of  the 
greater  minus  the  square  of  the  less  may  be  56 ;  and  the 
square  of  the  Iessj»/ws  ^  their  product  may  be  40. 

Ans.  9  and  5. 

"^r.J^^^^  ^''^  *^°  numbers,  such,  that  3  times  the 
square  of  the  greater  plus  twice  the  square  of  the  less  is  1 10  • 
and  half  their  product  j)/««  the  square  of  the  less  is  4.  What 
are  tne  numbers?*  ^«,.  6  and  1. 


i 


i 


CHAPTER  VI. 

ON   ARItHMKTICAL,    GEOMETRICAL,    AND    IIARMONJCAL 
PROGRESSIONS. 

1?'  ^f  ^A^^^^  o^  quantities  increase  or  decrease  by  the 
contmual  addition  or  swi^rae^/o/t  of  the  same  quantity,  then 
those  quantities  are  said  to  be  in  Arithmetical  Progression. 

llh^7n^  ^^""^  ''*"i°*^  °^  questions  relating  to  quadratic  equations 

hicn  contain  twn  nnV-nnwn  r,„^^Ht.i t»i       ..^    >  .     •       .  ^.    _      ' 


Which  contain  two  unknown  nnantiMV- 
tern*.  "^       "'"'"* 


•^-■jiutaicai  j-ror. 


I  or  42  11  7 
24 -=-3°' 4- 
ns  of  the  question  , 

48       48 


■==3-=16, 


<4=7. 


the  square  of  the 
iy  be  56 ;  and  the 
ly  be  40. 

Arts.  9  and  5. 

,  that  3  times  the 

of  the  less  is  110; 

fie  less  is  4.  What 

Ans.  6  and  1. 


IIARMONJCAL 

)r  decrease  by  the 
Tie  quantity,  then 
'Ml  Progression. 

quadratic  eqiiationa 


mmrnm^'.^^l^i^ 


AHITIIMETXCAL   PHOGKEobiO.,. 


J  nils  the  numbers    \    o.    n   a    r    n 

the  addition  of  1  ^o  ,\nh  f'     '  ''.'  ^'  '^^'-   (" ^icli  increa,^'  U 

^--  V^2  from  :^ach%u Lt  ;i  J^^"^^  '"''^r  V  che  .;./.,..; 

H'o  series  itself  bo  exprelerf  h     """"'?  ''i''"""^-  ""m  r  ,v 
"eco.-di„g  «s  rf  i.  po.mlltZ^ZT'"'  "  "  *-™"«/one; 

«"'e:,  1.  e.,  the  coefficient  of  j/^r'  ;"""=>'"■'«  it  is  ttr«, 

««.Jy  than  the  number  Vhieh  ITJ  '!,™  'f  """^'^  '«  '" 

'"  '^•^  »«■«•    Henee,  if  the  numhel    /t'^'""  "y'""' '"■»' 
be  denoted  bv  (,A   tV  '    ,u  """iibei-  of  tenns  in  the  serie, 

w"i  be  aH'L^'^i't:%iJzt  'r  r  ^^^  ^^^ 

^-  then  ^         "'''  "  "le  /ith  term  be  represcnte.t  by 

I?    ,   T..  ^=«+(«-iK 

i-x.  1.  Fjiid  the  50th  term  of  H,n       •      , 

TT  Luxixi  or  the  series  1   3  ^^  -y  ^ 

Ex.  2.  Find  the  12th  term  offk 

'"  '^^^"^  or  the  series  50  47  aa  a,^ 
Here  a=     50  1   •  /    rt\  ,  /,«        '      '  ^'  ^^- 

Ex.3.Fi„dthe25th~.e™„f;;,„,3_,_3_^j_^_^ 

'^"-        "     ^'^'"        '     IM,,f  " 

'>»f4n-l!;S'4| ""'"'"""•''  '"»-  ior  intennedftTte™!)- 
.    ^^ere  the  number  of  terms  i^  fi  ,.0      1       , 

!!!:!!::::^^;:^fs|~m^  «■ -™%.  ;h„  c^te„ns  to  b. 


V 


fi 


104 


ALGEBRA. 


!  It 


m 
m 


I II I 


')i; 


I     :    I  J 


I     ! 


I      i 


0=   1  /  But  a-f  (n  — 1)  d~l 
l=^4S[  .-.1+7^=43;    . 

n=  8  y  .-.  c/=0. 

And  the  means  required  are  7,  13,  19,  25,  31,  37. 

Ex.  6.   Find  7  arithmetic  means  between  3  and  59. 

Ans.  10,  17,  24,  31,  38,  45,  52. 

Ex.  7.    Find  8  arithmetic  means  between  4  and  67. 

Ex.  8.    Insert  9  arithmetic  means  between  9  and  1 09, 

74.  Let  a  be  the  Jirst  term  of  a  series  of  quantities  in 
arithmetic  progression,  d  the  common  difference,  n  the  num- 
ber of  terms,  I  the  last  term,  and  S  the  sum  of  the  series : 
Tiien 

^=a+(a+(/)  +  (a4-2(/)+ +/ 

and,, writing  this  series  in  a  reverse  order, 

Sz=l+{l-~d)-{-{l-2d)-\-    ...    -fa. 
These  two  equations  being  added  together,  there  results 

2  >Sr=(«+Z)  +  (a+/)4-(a+/)+  ---  +(a+0 
=(a+Z)Xw,  since  there  are  n  terms  j 


.n 


.-.  fc(a+/)^  - 


(1). 


Hence  it  appears  that  the  sum  of  the  series  is  equal  to  th« 
sum  of  the  first  and  last  terms  multiplied  by  half  the  number 
of  terms : 

And  since  l—a-^{r: — 1)  d; 

.'.S=.i2a-^{n-l)dl^ (2). 

From  this  equation,  any  three  of  the  four  quantities  a,  d^ 
n,  s,  being  given,  the  fourth  can  be  found. 

Ex.  1.  Find  the  sum  of  the  series  1,  3,  5,  7,  9,  11,  &c. 
continued  to  120  terms. 

Ilore  a=     n  ..,  g^  |  2a4.(n_l)  ,/  j-  x| 


d=z     2 


«=120J         =|2X1+(120-1)2[X-^. 

:=:(2+119x2)x60=240x60=14400 


f=43;    . 

J5,  31,  37. 

u  3  and  59. 

7,  24,  31,  38,  45,  52. 

sn  4  and  67. 

en  9  and  109. 

3ries  of  quantities  in 
iifference,  n  the  num,' 
be  sum  of  the  series: 

.  -  -    4-a. 
er,  there  results 
...  -  +(a+0 
^  terms; 

(1). 

e  series  is  equal  to  th« 
cd  by  half  the  number 


■'--  (2). 

e  four  quantities  cr,  d^ 
d. 

1,3,5,7,9,  11,  &c. 

44 

-1)2[X-^. 
0=240X60=14400 


ARITHMEIICAL   PP.OGRESSIOX.  I05 

^c^^tolo  SLV'^  «"-  ^^  ^^^  -ies  15,  11,  7,  3,^1,  ^5 


.=  -4l^^^         ^■'^"-')<'l^- 
=|2xI5+(20-l)x-4[x« 


flf=-  4 

«=     20J         =|2X^5+(20-l)x~4[x?? 

=  (30-.19x4)xl0  ^ 

=  (30-76)  X 10 

=  -40X10= -460 
14,  &/    ^"' ^^^«"-«^25  terms  ofthe  series  2,  5,  S   H 

34,V    ^'''  '''  --  o^^«  --  of  the  series^-  ^30' 

li^%f"^^^-""^«^^^^--o^theserie^r^t|! 
Herea=|.  ''■  ^=  ^^+{n—l)  dl!!, 

^=i      i         =|2XH(150-l)xj[i^ 

n=150  =/?4.149\  151        ^    ^ 

^  I3+T"/  '^=-3-X75=377.5. 

2it  &c.  ^"'  *'^  ^""^  «^  3^  terms  of  the  series  1,  1^,  2. 

^««.  280. 

p  PROBLEMS. 

term?  ""J  ^^^^^  ^0.     What  is  the /rii 


2& 


Here  A8'-„i 240 )  .     c      f«       ,  ^  » 

•^^"     ..    >S'=|2a+(n-l)rfl^ 

'^"■"  ^(  ^240=  J2a+(20-l)>   ./ J  ?( 

-=     20  J  =(2«-19x4)10         ^^ 

l24=2a-76; 
•■•  2«= 124+76=200. 

Hence  the  series  is  100,  96,  92,  <feo. 


Ill 


I  I 


IH 


'  fl 


f  i  t 


IN 


108 


ALGEBRA. 


Ptt'  2.  Tlic  «wm  of  an  arithmetic  series  13  H55,  the  ^r«J 
r -W  ,  .  eaid  the  nwnber  of  terms  30.  What  is  the  common 
nijferei    "^ 

IIer«  ^=14551       j  2.4  (n--l)rf  1 1*=5 

«=       ^r--.  |2x5[-(30-l)cfl^=1455 
n=     30  J  »t0+29ff)  15=1455; 

Dividing  both  sides  by  15,  10+29(f=97, 

29£/=87; 
.*.  c/=3. 
Hence  the  series  is  5,  8,  11,  14,  &c. 

Pbob.  3.  The  sum  of  an  arithmetic  series  is  567,  the^/'«/ 
Urm  7,  the  common  difference  2.    Find  the  number  of  terms. 

Here  5=507 1  .-.  since  J  2a+ (n-l)t/ j.  ^=5 


a=     7 
rf=     2J 


i2x7+(n-l)2l^=5G7 

7i'+Gn=567. 
Completing  the  square,  n'4-6n4-9=576. 

Extracting  the  square  root,   n4-3=,+24; 

.-,  n=21  or  -27. 

Prob.  4.  The  sum  of  an  arithmetic  series  is  950,  the  com- 
mon  difference  3,  and  number  of  terms  25.  What  is  the,/??-*/ 
term?  Ans.  2. 

Prob.  5.  The  sum  of  an  arithmetic  series  is  105,  theirs/ 
term  3,  and  the  number  of  terms  10.  What  is  the  common 
difference?  Ans.  3. 

Prob.  C.    The  sum  of  an  arithmetic  se:  '     '     4t'0,  first 
term  3.  and  common  diffe.^:dce  2.     What  >-  Hit   }.t  ."}>er  of  - 
terms  P  •         «  -a.^..  20.       | 

Prob.  7.  The  sum  of  an  arithmetic  series  is  54,  the  ^rst 
ifer»n  14.  and  common  difference  —2.     What  is  the  number  m 


tei    ' « i 


Ans.  9  or  6. 


jeries  I's  1455,  thajirsl 
What  is  the  common 


30 


1455 


5=1455; 

ic/=87; 
cf=3. 

JC. 


series  is  567,  the  Jirat 
the  number  of  terms. 


=  S 
=507 


567. 
►=570, 

1= +24 ; 
(=21  or  -27. 

series  is  950,  the  com' 
f  25.    What  is  the  first 
Ans.  2. 

series  is  105,  theirs/ 
What  is  the  common 
A71S.  3. 

;ic  seiii'-  '  ■  4*0,  Jirst 
''hat  '*•■■  •■;  ,  J>er  of 

}  series  is  54,  the  first 
What  is  the  number  ni 
Ans.  9  or  6. 


AHITILMETICAL   I'ROGKESSIOX.  j^? 

tfnrd,  and  so  on.    Jn  how  man t   ^        ^-^H  *''^'*^»  20  the 
journey's  end?  ""^"^  ^*.^=*  ^'^  he  arrive  at  his 

Uere  is  giver.  «=       30  i 

SZ    ~'iQly''^'''^'^^'^-^^^r  of  terms. 


.-.  |2x30+(n»l)x-2|^^=198, 

(31— «)n=i98^ 


w'-31«+ 


©'=? 


■198= 


109 


n- 


4  -  4 

31      ,13 

••"=2-±2-=22or9. 

pos1tL"?lir  of  rS  it"'^  r '"^^  ^--  'he  two 
the  traveller's  arrival  at  his  ifJrn/'  '''^  '^i^^-^^^  ;^^r^o^.  of 
that  if  the  proposed  serieiHs^r^''  \^  "^"^^  ^^^-^-' 
terms,  the  16th  term  will  htL;; '  '  *^?''  ^^  ^""^ed  to  22 
terms  will  be  «^^«^eW  L  wh  nh  "'^' !"^  '^^  ^^"^^'"'"g  ^^ 
traveller  on  the  16th 'day    anfh' ^ 

^^.c^^OM  during  the  six  days  foIIowin^"'^ ''u/^''  'P^''^^' 
hini  again,  at  the  end  ofX  9oTa^'  ^""^  ^'^'^  ^^^^  brin« 
^vhieh  he  was  at  the  end  of  the  l^h  '^'  '? '^^  ^^"^^  P«i»t  at 
place  whence  he  set  out  '^'  '''"•'  ^^^  "^"^^  from  the 

pthering  up  200  sSiS  pdntl"^  f!?'"  ^^'^  ^^^^'  h> 
-.^  2  feet  from  each  other  js^tl?^'!?.  k"''  ^'  '""''''^^^ 
^l^Oly  to  a  basket  stand  nffaf-f^iw     ^^^™g^^^«h  stone 

[he  first  stone,  and  C^X^Zt^^T^  ^"^  ^"^'^^  ^^^^ 
basket  stands?  ^^"^^  '^^^^  the  spot  where  fh^ 


3  jj 


ft: 


t 


/* 


% 


108 


ALGEBRA. 


i-ii 


M  III 


'//Sb."-  "'^"'')'  "«-»''(#•--  2  feet,  and  «„„J„ 
Here  a=  60 )      «      f  n      ». 

«=200)        =(120+398)X100. 

=518x100=51800  feet, 
ffenee  the  distance  required=103"600=  I'q    .""'i""'       .'"'• 

What  i,  ««-a^:;'eSsf' ' '"'  '"^f 5?ro-- 

the  .a4,  aifd  so  o^'  "wfft' ml'^  H"?  '*™>-«4" 
ohaHt,at.he,endofihe^\™"^^t..''lcrir6Vi''' 

before  B;  B  follows  him  Tl  !^  I  '"""''^  ""'•  20  mhiutos 
«  the  second,  7  the  Mi™  and  so?  '  f  1"""^^  ""'>"'  hour 
B  overtake  A»  '         '°  ""•   '"  '"'»'  many  hours  «-i|l 

^M.  in  8  hours. 

arithr«e' proIL'slo^'  LT"^  """"'f  "^  1™«ities  u, 
whose  «»»  is  ela  to  ll?!-  T'.'  ''^^'•^««  ^^  2.  and 
if  13  be  added  to  heZ„!;r''"'°'';T""'"-'  """-"'ver; 
by  the  numiJofV^,  S  Lto«  ";  ""?,?  ?'^  ^-^  ''<'  divided 
«™>.    What  aro^Smberrr       """  ''^  "l"*' '»  *«>« 

Let  the/r«<  term=x,  } 
and  iVb.   0/  tenns^y;  ]  ^nen  the  wcowrf  term  will  be  a'+2. 

'^  '''  ^^^''"^^'^'^  ^^^+(^^^X|,  substitute  .for  .,2  for  ft, 

and.for.,  and  it  becomes  2l+(^riy3^|(^,y..__        '    " 
li>r  the  swm  of  the  scries. 

By  the  problem,  a^y+y^^y^^^^  ^^  ^^^^^^ 

unci  — : — -:.  '-!-., 


GEOMETRIC   PROGRESSIOX. 


et,  and  numitef 


109 


ftirbngi, 


feet. 

G40 


?ave  1  shilling 
ird,  and  so  on. 
w.  £110  95. 

giving  away  a 
hree  farthings 
disposed  of  in 
^  lis.  6fd 

of  6  miles  an 
1  20  minufes 
hQ^first  hour, 
17  hours  \vill 
n  8  hours. 

[uantities  in 
«  is  2,  and 
;  moreover, 
1  be  divided 
1  to  thojirsi 


2. 


•r  a,  2  for  b, 


Hence,  ^^?=;r,  or  ;t«-8x=-15; 

.*.  «'-8^+lC  =  16-15=l, 

anda:--4=  +  l;...;r='5or,1, 

Fmm  which  it  appears  that  there  are  ^^^J'sets  of  numt,. 

...•^"'^'':-  ^"^^    ^'''''''   '^  ^  ^^^^a^n  number  of  quantities  in 
arithmetic  progression,  whose >.^  term  is  2    and  wW  Lm 

Am.  2,  5,  8,  11,  14. 

ON  OEOMETHIO    PROORESSION. 

ra.    If  a  series  of  quantities  increase  or  decrease  hv  .!,„ 
contnmal  ,«,,//,>&„&„  or  division  by  tl,e  same  qum.Utv  then 

continual  ^^.-Wiif '  ^AfanlVet^^^^^^^        Y  '•>; 
m,dl,pl,cal,on  by  J),  are  in  Geometrical  Progression 

76.  In  general,  if  a  represents  the  first  term  of  sncl,  . 
series  and  r  the  common  mulliple  or  ratio  then  mw  h! 
series  itself  be  represented  by  a  or  or'  n^'  „Ta.  ""y  '''« 
will  evidently  be  aL-„«..«„/o"r'  ZZ^i^;  'ser  es  „e;,„:S 

eri;s"u,e1l;':f '••"^  ''^™^^'--^'™''°'' '" "» °'^°g"  g 


""'"  "  "'  geoirietncul  progression  found  f 


~5 
tiie  common 


"'  -mmiikm  ei-ii),, 


no 

ALGEBRA. 

iJere  the  common  ratio  -?-o 

"~1 — ^• 

^  2,  rind  the  common  ratio  of  the  series  ^  ^    ^    . 

3'  <>■  27'  ^^ 
I"  this  example  the  common  ratio  =  ^^2_2 

9'3-3' 
•^x.  3.  imd  the  commo,,  ratio  in  the  series  5  1  ^    ^    ^ 

TO.    Lot  S  be  the  sum  of  the  aeries  a,  „'  a^  J  „ 
Mu]t,p„  .he  eWtion  b,  .,  a„a  itbeeot;      "  '      =  ^^ 

S"i'-ttho.,,.e,„..„,„„.:,:-j-=^ 


•••'~a=.^^^,orO.-.l)^^„,.. 


-a; 


and  therefore,  S=^^^~:Zf^ 

_«  the  convenienee  of  ea,cu,ation,  .he-efore,  it  is  better 
'"  *"  '"'"'  '^  "-"-pose  the  equation  into  S  -  «-'"■"  t 
'"i'ltiplj-ing  the  numerator  and  dnn„    •    .       ~  ^^^'  ^ 
— -  hv  -1  denominator  of  the  fraction 

r—1      y       ^' 

™-    "■  '  bo  the  last  term  of  „  series  of  this  Icind  „ 

W"tion.lheref„re,if„„y,h_^/f7'^,    '-l'    ^""^  *^'» 
l,  ke  given,  the  fourth  maVbefoun  *"/"'"'  '»"'""■«-  ^'  «.  ^ 

78.  Whnt  i«  fi,«  „ ^      7     !  ~~ ■ — — ___ 

^  «'^«'"«trieal  K^gr^^i^t"  '"^  "'«  «»'»  a^" «  terms  of  u  series  of  „„n,bo« 


cal 

progression 

4 

8 

2 
3^ 

2 

3 
5' 

0 
25'  *^^- 

A 

3 
ns.  r. 
o 

^, 

^c,  then 

=  S. 

ar" 

=rS. 

we 

have, 

^rs  are  Jess 
it  is  better 

he  fraction 


kind,  then 
>om  this 
?s  ;S;  a,  r, 


>nnimb«n 


GEOMETRIC  PKOGRESSIOX. 


HI 


Ex.1. 


Find  the  sum  of  the  series  1,  3,  9,  27,  &c.  to  12  terms. 


Here  a  =  1 
r=  3 

«=12J 


.^_^"^2^_  1X3*^-1 


r-1  3-1 

_8P-1 

""     2     • 
531441-1     531440 
■        2 =~"2 — ='*^5'^20. 

Ex.  2.  ^- 

Find  the  sum  often  terms  of  the  series  l4-?4.^-.   ®    ^ 

3^9"^27'     ^ 


a..-  1 

2 

r_=    - 

3 

f?=rlO 


3 


Now  L2)'"=2L»^  ^4 


■■-sr- 


3'«  "59049 ' 

1024^58025 
59049  ~59049' 
3x58025     174075 


and  S=z 

59049         59049  * 

81^^^-    ^"'^  '^''  ^""^  "^  ^  '''"^^  ^f  ^h-  series,  1,  3,  9,  07 

Ex.  4.  Imd  the  sum  of  1,  2,  4,  8,  16,  &c.  to  14  terms. 

,    ,    ,  -^ns.  1C3S3. 

Ex.  6.  Find  the  sum  of  1,  -,  -  -1  &«  m  ft  f 

3  9'  'n'      •  '^  ^  terms. 


'3' 9' 27' 

.      3280 
Ana. . 

Ex.  6.  Find  three  geometric  moans  between  2  and^f 
Herea=  2)  And  ar-^'^/ 


/=32 


I 


•  0..4 


2r*  -32, 

:JG. 


And  the  means  required  are  4  8  10, 


rii 


112 


ALGEBRA. 


iiii ! 


:ii:|iiF 


11 


^.  7.  .Find  two  geometric  meana  between  4  and  256. 

P     «    -P-  J   ,  *  ^^^'  ^^  ^^^  64. 

J^x.  a  Find  three  geometric  means  between  |  and  9. 

p^  0   -p.  ,  "^^^-  i»  1,  3. 

T  «?    tSf,"^^t"«  "lean  between  a  and  /. 

Ihen  a,  x,  /,  are  three  terms  in  geometric  progression. 

and  -=  _ 
a     X 
or  x^=zal 

.'.xz=y^ 
Ex.  10.  What  is  the  geometric  mean  between  16  and  64 1 

Ex.  11.  Jr^ert  four  geometric  means  between  -^  and'sT 

Atis.  1,  3,  9,  27. 

PROBLEMS. 

squares  to  21.  ^        *^  ^ »  ^"^  ^^^^  «wm  o/M«V 

I'-fc  p  ^,  :^y,  be  the  numbers.  Then  by  the  problem, 

X 

X* 

'~,+x'+xy=.21 .  .  (2) 


Trom  equation  (1),     x{l+l^.y\  ^^ 

'•  Vsquanng,  -'g+?+3+2y+y]:=49 

.%  by  subtra«tir.ii,    a:'^^^^2+2y\  ~I^ 

or  14z=28 

*.  X=:    2. 


4  and  256. 

'S.  16  and  64. 

>n  I  and  9. 
Am.  ^,  1,  3. 

ndl. 

'  progression, 


en  16  and  64? 
Ans.  32 

1  }  and  81. 
1,  3,  9,  27. 


5  progression, 
3  sum  of  their 

problem, 


GEOMETRIC  PROGRESSrOJf. 
Tliis  value  of  a;  being  inserted  in  (1), 


us 


5±3    ^      1 
Hence,  the  numbers  a: e  1,  2,  4;  or  4  2  1. 


.•.y= 


Ans.  2,  4,  8 ;  or  8,  4,  2. 
nhosT";^-  islr^nll'/^''  """J^f '  '"  S^^"^^^'«  profession 

Ans.  3,  6,  12. 
^>,o  .  °°'  ^;P°If  ''''^  **'''^^  numbers  in  geometric  procrression 
's  100.  What  are  the  numbers?  Ans.  2,  10,  50. 

Prod.  5.  There  are  three  numbers  in  geometric  proffressio,. 

atrr„'^er  'r^  *"  ^^  o'  *^>-™^  'j«°i«S 

-4/js.  1,  5,  25. 


I 


OS   THE   SUMMATION  OF  AN  INFINITE   SERIES   OF   FRACTIONS    ,H 

ThTvaZ  oTr """"''   ^•"'   "^    ™--™0O0F  Z,N„ 
THE  VALUE  OF  CIRCULATINa  DECIMALS. 

«nJ?'    ^"^  ^^"''''^^  expression  for  the  sum  of  a  ffcometi-ln 
sct^es^hose  common  ratio  (r)  is  a  fraction,    is  flTl^) 

'^'=T=7"'    ^"PPOse  now  n  to  be  indefinitely  great,  then 
'••  ('•  being  a  proper  fraction)  will  be  ii  defin  tely  ma//,»  »j 

*  When  r  is  a  proper  fraction,  it  is  evident  that  r"  dccrea^n  as  » 
increase:   let  r=-  for  instance,  then  r^=l-  ..--J=  ^-__I      . 


114 


;l  I 


*     'i: 


■  I  fl 


ALGELRA. 


tt^CraTor'::^^^^^^^^^^^  ^-^^  -Peot  to  a  in 

v^hen  the  number  of  terms  is  infmite  is-^  ^ 

'     1-/ 

Ex.  1. 

Find  thesum  of  the  series  l+I  +  U^  a,,      ,  •  ^  . 
Here  a=:l  )  _  , 

2 
Ex.  2. 

Findtho  value.ofl-L-ij  _L,.        _,  .   , 


Ex.  3.  Pind  the  value  of  H-i.  -,-1  .  ±  .  c        , .  ^  . 

Ex.  4.  Find  the  valueof  l+l+i.  .  27  ^  .  ^ 

Ex.  5.  Find  the  value  of  1+1+ A  .  ...     .  •  t""''  ^' 

bers   composing   whiIhL<,''"^f^"^  ^'"■"^«^^'  ^^e  num. 
_____^_»J^^^^  progressions,   whose 

80.   W»iD»  to  *u .        \  ~  '        ■  ~ — ■ ■ 

uun.bcr  ortenua^;  alter"  '"''"'  '""'  of  ^geometric  series,  when  th. 


■ha4i 


respect  to  a  in 
[ing  the  valua 
'>S' approaches, 


infinitum. 


'  infinitum. 
Ans,^, 

infinitum, 
dna.  4, 
'■nitum, 

Ins.  ^. 

tJous  me. 
the  11  um- 
»s,  whose 

'i  when  thi      * 


GEOMETRIC  PROGEESSIOX. 


llo 


1        1 


1 


common  ratios  are  —      ~         *       »  •,. 

10'  100'  1000'       '  *c^<^rding  to  the  num. 
ber  of  factors  contained  in  the  repeating  decimal. 

^-  1 

rind  the  value  of  the  circulating  decimal  .33333  &c    llils 
d^ecmal  is  represented  bj  the  geometric  series        ' 

TO'^'IOO'^IOOO"^*^''-'  ^^^'«  >•«'  '^^»*  is  A  and  com»ao« 
1  ^" 

10 

Hence  a= — . 
10' 

-2 

**~10' 


.-.  aS'= 


3 

10 


1-r 


1— L  ^^-^ 

10 


9~3 


Ex.2.  Find  the  value  of  .32323232,  (fee.  at/ m> //«;;,. 

32 


Here  a= . 

100' 


r= 


1 
100' 


1— r 


32        32 


1 


1       100-1  ~99* 


100 


Ex.  a.  Find  th«  value  of  .713333,  &c.  ad  infinitum. 

The   series  ^of  fractions  representing  the   value    of  this 

decimal  are  — +  (geomet-io  series)  --1-+-J_  .     &. 
100  ^  lOQO^lOOOO  ^   ^^ 


100"^   • 


Here 


,.__ 


.-.  S= 


3 

1000 


3 


1 


J 1     1000-100~900~306' 

10 


Iknce  the  value  of  the  decimal=/~+^)Zi4.  .1.^214 
\tvy  X'^OO      /iOO    3oo"^a0(j 


107 
150' 


116 


ALGEBRA. 


^^'  4.  Find  the  vaJue  of  .8134^4^4  a 

2re  a=r-?f    1  -^^^4^434,  &c.  ad  injinittm. 


Here  a=Jl_  1 
10000 

\ 


1 


34 


.^=--l_^_iO00O^ 34 

'"'=100     I  ^~'''    1— i-    ^^^o^o-ioo'^y^ 

^  100 


And  value  of  the  decimal  =ii  .  e     81    ,    ^ 
^x.5..Findthevalueof.77777,<S.c.a.f/.>,V.n. 


100  100^9900-i)900- 


Ex.6. 


Ex.7 


EjT.a 


Ex.  9. 


«        « 


9 


.232323  &c.  ac/  m//»7«m. 


M 


Ans. 


23 

99' 


M 


.83333,  &c.  arf  ^V/^^Vw;;^. 

Ans.  --. 

•'^l4l4I4,&c.a«?e;y?;,,V..m. 

.      707 
Ans. 


u 


•956806.  &c.<,rf  ,■„>,•,„„, 


Ans. 


990* 

287 

Let  S=:     .813434 
.-.  10000  *^=8134.3434 
a^dj!00^__8i.3434  .'  .*  *  .* 
.-.  9900  >S=8053        " 

.   <._8053 
'  *        9900'  ^^  ^^^"^-e. 

-"V  "-ves  over  ^  a  mile  the  ^mw"  '"-■^"^^Vg  cause,  it 

*m;,i«r  second,  |  the  M,Vd; 


fnjiniium.  \ 

f 34 

-lOQ-yyoo 

8053 

A      23 
99 


im. 


ns. 


Q' 


itum, 
707 
990' 


ns. 


im. 


ns. 


287 
300* 

found  1 


-ause,  it 


HAKMOXIO  PROGRESSION. 


117 


and  so  on  Show  that,  according  to  this  law  of  motion,  the 
body,  though  it  move  on  to  all  eternity,  will  never  pass  over 
a  space  greater  than  2  miles. 

ON  HARMONIC  PROGRESSION. 

81.  A  series  of  quantities,  whose  reciprocals  are  in  arith. 

luetic  progression,  are  said  to  be  in  Harmonic  Pronression. 

Ihus  the  numbers  2,  3,  6,  are  in  harmonic  progression,  since 

heir  reciprocals  ^,  ^,  J,  are  in  arithmetic  progression  I  -4 

being  the  common  difference).  ^     ' 

Ex.  1.  Find  a  harmonic  mean  between  1  and  — - 

3* 
Let  X  be  the  mean  required  : 

^^"  ^'  T'  "2"'  ^^^  ^"  arithmetic  progression, 

Andi-l=i_i 

X  2      X 

X        ^2 
5 


.*.  af= 


2 
4 

5* 


Ex.  2.  Find  a  third  number  to  be  in  harmonic  pro^re«8loi 
mth  6  and  4.  i     o        ^ 

Let  X  be  the  number  required : 


Tnen  — ,  ~,  —,  are  in  arithmetic  progression. 


All  1 

And  - — --: 
4      6 

1 

* 

X 


1 


1_ 

X      4 

-i-i. 
2       6 

1— i 
1 


111 


1'-' 

r.l 
I:  if 


ar=3 


118 


V  i 


ALGEBRA. 


rhe  re.,rocak  of  9  and  3  arc  I  and  ^    .,  •  k         T 
and  last  term  of  an  arithmer      '  ^'  '''  '^'•^^'■^' 

amr^Uc  means  a^tf  r^inSeTt'  V^^^  ^^^^  3 
according  to  Art.  73—  "^^erted.    We  have  therefore. 


3 

^=5j 


And  a+(n— Ijc/:--/^ 


3      9 


.*.  d=z 


3^ 

'9 

9 

18' 


9 


Hence  i  1    ^  l^' 

'^6'  9 '18'-^  the  arithmetic  ..«.,  to  be  inserted 
between  ^  and  ^,  and  therefore  tkeir  reciprocals  6    ^     ^^ 
are    .three  harm^^^     --  -.uired.  '  ^'  ^' 

•  '•  ^"'  ^  ^^^---  —  between  12  and  C. 
Ex.  5.  The  numbers  4  «n,i  p        .  ^»»-  §• 

progression;  find  a  third  te™       ^^  '"°  '"™=  »''''  ha™onic 

E^-O.Eindt.o  harmonic  n.e.„s  Ween  S4  .:!":;" 
E..7.InseruhreeharmoniemeansheJ::r5:;fdf 
S2.    Let  a.  i    c    t/   ^    ^       T.  2 ' '''  4  • 

*«-».p4re:sio;:;hli'i%"r7°'"^™"«"-" 

«  '  b'  c  '  (/'  7'  ^C"  are  in  arith 


and  3. 

are  thejlrst 

een  M-hlch  3 
}  therefore. 


inserted 
'  2'    5^' 


ns.  8. 

trmoiiic 
'.  12. 

56. 

163. 

is. 
15 

I* 

ties  in 

arith, 


HARMONIC  PROGRESSION. 


119 


a^ 


From  (1) 


-i-i-i-  1 

b       a       c       b 

..(I) 

•  ■  -  (2) 

^      c      e       d 

(3) 

&c.      =      &c. 

a--6      h—c 
ab           c 

a—h     b—c 

a           c 

a     a—b 


c     b—c' 
o    converting  this  equation  into  a  proportion, 

a:c::a — b:b~c 
Similarly  from  (2)     b:d::b^c:c-d 
(3)     cie-.-.c^did—s 
and  so  on  for  any  number  of  quantities. . 

These  proportions  are  frequently  assumed  a/fh*.  /.w   •.• 
to   quantities   in   Aamo^^zc   progression    nnf,!.     'Y'''^!!''' 
expressed  in  words-— if  anv//,;.^T!l..     "^^>'  ^^  *^"« 
gression  be  taken    M.  f^,       /  quantities  in  Aa,v«on/c  pro- 


a«=jy;  .'.a—b» 


c*=b^',  .'.c=bi 


(1). 


r2\. 


By  multiplying  (1)  by  (2)  ac=b> 


!  + 


■J,  • ;  I 


■I 

ii 


i-i;!i 


-s?.«^^M«e.t*-»n*f^»^t«.v;'J^ 


I       I    I 


$ 


=1 


V        Vl 


120 

ALOEBKA. 
l^ut  by  geometric  progression  ac=A» 


and  .-.  2=il+il 


y    X     z 
or  i-i=i_l 

y  X  Z  y' 


,  CHAPTER    VII. 

ON  PERMUTATIONS  AND  COMBINATIONS. 

83.    By  Permutations  are  meanf  t]io  «.,    i, 
«-lHch  any  quantities,  n  i  c  Te  V  ""^^^  ^^  ^'''«'^^^' 

spect  to  theirorder,  when  takon'/      '  T  ""^'^""^  ^'^h  m 

f ,  «?.,  are  the  difre;ent  pe^ml^e^^^^^^  ^«'  ^^'  <  ^«. 

^  ,<•,  c/,  when  taken  two  aTdtZfLf^^     ^T  ^"^"tities  a, 

7^,  ci.,  of  the  ^Am  quantifies  Vr/.'  "*'',  "^•^'  *^^'  *-« 
Mre.  together,  &c.,  &j"^"^'^*'^  «'  *'  <^.  when  taken  ^Are,  „  J 

'^'^m:^z:z:t^  th.,  hy 

(«-!)  be  substituted  for ;.  in  tho  l^^Vo^-T''?''''''^'-  *^'^''  '^ 
of  permutations  of  n-^l  Z!t  ft  ^'^f  ^^'  ^^^"  ^he  number 
will  be  (.-1)  (,^,)  ^  jJj^^X  '±Vr.f!!'  ^-  together 
->  c,  a,  ^  O.C.,  taken  t^o  and  two\og^^  a^e'^^J^;:!!^; 


fsiWi,  ^r.^Agmm^^,*^^me'—-  ■. 


PERMUTATIONS. 


121 


f  changet 
»  with  m 
her,  three 
•*,  cf/,  </«, 
m titles  a, 
<'>'0'c,  Acer, 
?Aree  and 


then,  by 
ations  in 
B(n--l) 
0^,  ^,  &o. 
■he  form 
e. ;  i.  e. 


d  two 


iS 


)gGt  her, 
For  if 
lumbei' 
3get  her 
/ions  of 


quantities  a,  6,  c,  d  e^,  &c,  taken  //.r..  fl;.rf  three  together  ir 
which  a  may  stand  first    for  the  same  reason  there  a?e  (n-U 

il;i         ^'  ""'"*''  ;  '''  '^^  permutations  of  his  kind  viU 
therefore  amount  to  .  (/i— 1)  (w— 2). 

r  ^fetHer.  ^'''^  '^''  """'*''  '>/;>^m./a^/o„,  of  n  things  taken 
By  Arts.  85  and  8(5— 
llie  No.  taken  /m;o  together  —  ;i  («- 1)  --" 

"  "      '/'^^^      "        =n  (n-1)  (,,«o) 

Sm.dar]y    "     four       «        =n  (n-1)  (n>2)  (.-3) 

supposed  'tX'ttLZ'^^t  is  ff ^;h^  P""f  ^^  ^'^^^•^'  '^ 

taZs  of.  things  rcf  !('^t  tXi;  t  ^rgetw  r-- 

n  (,.-1)  (,,^2) (/.-r+2) 

will  be  by  putting  .-I  ^ :):: ^X^^;:-^^^^^^-' 

{n-l){n-2) («~r+l) 

Now,^if  a  be  placed  before  each  of  these  permutations,  t^'^ore 

{n-l){n-2) («-r+l) 

ft1s"ciraftLl'e>"-ir''"ry  *^^^^^"^'  ^"  ^-^'^h  -  ^^-^d^/'--^^. 
ic  IS  Clear  tnat  there  will  m  like  manner  be  the  same  nnm)u.n 

here  arl  IS  *'  ^'h?  *!•'  ^'»"d  mr.o&./y /„, ,.  »,„,  „, 

.vel_y_sta„d  first,,  that  is,  „  times^ri)'' [t^^g]  "^.'•f';'; 

'or«(„_l)(„_2) („_,.+  ,) 

nJ!  •       *"\''<*™  P>-oved,  that  if  the  law  by  whioh  the  c^ 

^TI^JH^l  ".'"""-.of  permutations  of  ^,  thln's  t^kt!; 

-    "-„„_.:ci   13  iouna,  oe  true,  it  is  also  true  for  tho  nov* 

uperjor  number,  or  when  „  tiling,  ,,«  tak  „  r,i::,      ! 

but  the  law  of  .he  expression  ha?  been  found  to  l,f,lj  fi',' 


5        XI 

1:1 


p»»«(f^.^«,»««^^jf^,,,,.V.^^^^^ 


122 


ALGEBRA. 


m    m 


taken  four  togetfer  k  U  t.^  T  '°«f ''^'•'  '"'^  '^  "■"«  «to 
and  si  on  fo?any  number  no?  „  "/""T  ''^™-^^'^  '"S'^'her, 
taken  together    ^  '  «'''''"*'■  "">"  »>  »Weh  may  be 

mathematical  sconces  ®      ^''""  "''l>°'-t»»'»  in  the 

titicI'al^;=%Ln  (Itn'-^'tr  "^"f  ""  *^  •)"""- 

mutation,  which  might  be' fo;m;dV,.^^,'^,*,^""™''''•  "''  P" 
the  word  •.«,.„...  aVflxS/xTxiLlar  '°'"''°''"" 

thefit^t\^trbi::tutSrrTr"'''-°^'''"-. 

mutations  by  the  number Tf  1?      .'     "^^hole  number  of  per. 

arisen  if  .^/j;j:ZsXScTi;,st::d"Si"°^^ 

of  the  same  letter.  Thus  if  the  samrilft  1  Z,''^  repetition 
then  we  must  divide  by  2x  1  ?/,•!"■  m"  °°™'' ""'«' 
.nust  divide  by  3X2X I ;  if  p  t  ,ncs  bt f  of  ''""  "T'  ^ 
any  other  letter  which  mnO  „„i     ^        ,•  •  •■?'!  »"''  ^o  for 

the^eneral  e.xpre2"n  forV  ,  IC'o^f  ?"  Tt^  ^'^™» 
thmgs.  of  Which  there  are  ,  l^'T^lX  IZmZ'  "' o' 
another,  &e.,  &e    is  ^ii!iriijfcl2H!ii:3)  •  .  2.1 

the  permutations  which  may  be  formPfl  Z.r  ,i   \  .. 
posing  the  word  "m.m..."iince  .TccmVl;     '  ^'""''  T^^" 
8.7.6.5.4.3.2.1  '''*'^'  ^  ^"■'''')  '^^'^ 

1.2.3.  ><T2~~^^^'^- 

can\^;  U'":;^VlTat^Sr,r^^^^^^     «"'«'■ 
The  number=l  X2x3x4x5xO=T20. 

-pS'biHr'"'  ""'  """"''"  °f '''""S-  -»"<=•■  can  be  rm. 
The  No.  o.  <">a„ges=lx2x3x4x5xcx7x8=40320. 


^^^^mm. 


w  together 
taken  three 
smonstratcd 
*  true  when 
i-'e  together, 
ich  may  be 

nonstmiive 
mce  in  the 

the  quan- 
'  them  will 
>er  of  per 
composing 

I*  of  times, 
^erofper- 
ould  liave 
repetition 
cur  twice^ 
'hrice,  we 
nd  so  for 
.    Hence 
ions  of  n 
her,  q  of 

.     Thus 

Bi's  com- 
oice)  are 


ts  which 

be  rung 
0320, 


COMBINATIOXS.  ^23 

Jl\I:t  "  ""S^  °f ''"'■^'-'  «<"»"-■  how  ™a,.y  u.jnaU 
The  number  of  signals,  when  the/aj,  „e  taken- 


Singly,  are 

Two  together  =5.4    .    . 

Three  .    .    .  =5.4.3    -    . 

Four    .    .    .  =5.4.3.2     - 

Five     .    .    .  =5.4.3.2.1 


=  5 
=  20 
=  60 
=  120 
=  120 


.'.  the  total  number  of  signals  =  325 

ietfeS,tuti  ariiiT""""'"'  ■=»  ''''''7:^:^^'' 

^I'f  ^Ai  ^^^  ^"f"^  permutations  can  be  formed  out  of  th.. 

Xn  af  S  ""'  ^''"^^'^  '-^-'™'^.  ^.  thetSf  j!^ 

Aris.  2o20  and  1680. 

ON  COMBINATIONS. 

w|^!h  ?j„ttr:7onrofr;t,rtr/f t"' 

are  the  ««6»„&„  which  ean  hef^tl/lfJl.'/' 
quantities  a,  i,  c,  d,  tal<en  too  and  Jo  toUther  at  { J 
icd,  bed,  the  combinations  which  may  bo  formed'„nr;,f-  ,j  ' 
same  quantities,  when  talcen  three  and  iree  toXll   &c' 

i.n..;e.iiately  deduce  the  theo^^t  ' &  ui  '  th  TmtvTf 
r»«4»,a  .o,«  of,,  things  tal<cn  in  the  same  manner     H'oMhi 

IT'TIT  "'  :  '^''T  "'''™  """  «'"'  '-together  Ling  , 
t«  - 1 ),  and  as  each  eomb,m,lion  admits  of  as  many  »ot«,,2„ 
us  mnjr  bo  made  by  two  things  (which  is  aTlf  tlTnun      r 

li  vXd  by  rreTnumb^""'  r'"  *?-""""'^^  'Ut^r,;': 

uiv.aca  By  ^,  I.  e.  the  number  of  comSwarions  of »  things  takes 

.'"'0  and  two  toccther  is  "  ^"     ^)      v^^  .1 

i.uj,i.im,i  IS        ^ ^     J,  or  the  same  reason,  th« 


79  I 

ALGEBRA. 

co^nbinations  of  n  things,  taken  three  and  three  together,  n.usi 
be  equal  to  !Lferi)_(!i-2)         ^  . 

1.2.3         ;  anci  m  general,  the  combhia. 

'S^lU^S^^^  together  »„.  bo  e,u„,  to 

The  number  -8X7x6x5x4 
P     o  ..r.     .  1X2X3X4X5-^^- 

be t;nL^Sfof  6  oT'''  ?r ^'.'  "^  combmations  which  can 
lurinca  out  ol  6  colours  taken  in  every  possible  wav  ? 
No.  of  combinations  when  the  colours  are  taken-^  ^ 
1  at  a  time 


4 
5 

G 


_6.5 

~1.2  -    -    - 

6.5.4 

1.2.3    '    "    " 
_  6^.4. 3 
"1.2.3.4    ■    ' 
.6.5.4^3^ 
"1.2.3.4.5    " 
.C-5^.jk2J. 
"1.2.3.4,5.6 


=    6 
=  15 


=  20 
=  15 
=  6 
=    1 


Hence  the  total  number  =  63 


_Ex._3.  Find  how  many  different  combinations  of  8  loftn,, 
i^ken  m  every  possible  way,  can  be  made.  ^^'"' 

%♦  Several  other  useful  and  interesting  subiects  of  ^Z'  ^^^' 
character  yet  remain  to  bo  treate-l^ofTh-E^litn^!     '""*"'"'" 
for  puMacafon  a  Second  Part,  wh.da  will  elnbrltlULa^"?* 


gether,  musrt 
he  combina- 
be  equal  to 


hich  can  b« 


is  which  can 
!  way? 


APPENDIX. 


MinCBAL    PRINCIPLES,     PROPERTIES     OF    NUMBERS     AVn    otu-t, 


8  letters, 

s.  255. 

ilementary 

preparing 

e  subjecU 


ON   THE   DIFFERENT    KINDS    OF   NUMBERS, 

1.  A  number  expresses  single  unif«  nr   na^fa   .e       •     . 
m-.^,  or  at  the  ,,a,„/timo,  unit? and  pXfSs  °'  "  """'' 

By  unit  we  understand  a  whole  1  ai-d  hv  Tzt,    e 
c-  ^fraclion,  all  that  is  below  the  vkue  o/l"  ^"'^  "'  "  ™"' 

2.  The  number  which  expresses  units  only  is  called  a  whnh 

uiL.^  hucn  as    1,2,3,4,  is  called    an   abstract   mimlwr 

J;  :»r."''CTh";r:^d^wf.hi «« '5  "7  '^vV-''^/"; 

an  odd  number.  '    '  ^'  ^'  ^^  ^'  '«  called 

V^c,  is  called  a  comj./.a:  numberr  ^~  ''  "'  "'^  '^''  '"'  ^•■*'  ^^'  ^•'^ 

1]* 


126 


APPENDIX. 


The  exact  divisors  of  a  complex  number  (or  multiple)  ar« 
called  submultiples.  Thus,  foi  example,  1,  2,  3,  4,  6,  8  and 
12,  are  submultiples  of  24. 


¥    I 


ON    THB    FOUR    RULES    OF   ARITHMETIC. 

6.  By  Addition  we  add  two  or  more  numbers  so  as  to 
make  but  one.     The  result  is  called  sum  or  total. 

By  Subtraction  we  cut  off  a  number  from  another  num. 

o    ??^  ^  ^^"^*^  ^^  ^'^^^^^  ^^'^  ^^"^^^^(^^r,  surplus,  or  deference. 

By  Multiplication  we  take  a  number  called  multlvlicand 
as  many  times  as  there  are  units  in  another  number  called  the 
multiplier.  The  result  is  named  the  jororfe^c^.  The  multipli- 
cand and  the  multiplier  are  called  the  two  factors  of  the 
product.  * 

By  Division  we  ascertain  how  many  times  a  number  called 
the  divisor  is  contained  in  another  called  the  dividend.  The 
result  IS  named  the  quotient.  The  dividend  and  the  divisor 
are  na:ned  the  two  terms. 

7.  The  unit  neither  multiplies  nor  divides. 

8.  To  multiply  by  a  number  less  than  the  unit  is  to  di 
nuimh  the  number  multiplied  ;  hence  it  follows  that  to  mulli- 
pi'/  IS  not  always  to  increase. 

To  divide  by  a  number  less  than  the  unit,  is  to  increase  the 
number  divided;  hence  it  is  that  to  divide  is  not  alwavs  to 
air.uiisk.  "^ 

9.  Of  the  two  factors  of  a  product,  the  one  being  multi- 
pbcd  and  the  other  divided  by  a  like  number,  the  result  re- 
mams  the  -lame. 

10  The  two  terms  of  a  divisor  being  multiplied  or  divided 
by  a  like  number,  the  quotient  remains  the  same. 

11.  The  general  product  of  several  numbers  is  always  the 
same  in  whatever  order  they  are  multiplied. 

12.  A  quantity  multiplied  or  divided  by  a  number  give" 
the  same  product  or  the  same  quotient  mul'tiplied  or  divided 
successively  by  the  factors  of  that  number. 

13.  If  the  quotient  be  greater  than  the  unit  thp.  a\rr\Ai^i'4 
IS  greater  than  the  divisor,  and  vice  versa.  If  it  be  the  unit 
the  divisor  is  eoual  to  the  dividend. 


Ml 


idMh 


APPENDIX. 


dtiple)  are 
t,  6,  8,  and 


127 


1  so  as  tc 

ther  num* 
difference. 

iltlplicand 
called  the 

3  multipli- 

rs  of  the 

ber  called 
^.end.  The 
le  divisor 


is  to  di 
;  to  multi- 

3rease  the 
ilways  to 

ng  multi- 
result  re- 

r  divided 

ways  the 


)er,  gives 
r  divided 

the  unit, 


14    If  the  product  of  two  factors  be  less  than  their  sum 
It  IS  bpcausc  that  one  of  them  is  necessarily  the  unit. 

15.  To  d<mhle  treble,  quadruple,  centriple,  &,c.,  ii  nmuhcr, 
IS  to  multiply  It  by  2,  3,  4,  100,  &c. 

10.  A  quantity  multiplied  and  divided  in  turn  by  a  Hke^^ 
^nmber,  becomes  again  what  it  M^as  at  first.     In   this  case  ' 

Ind  dividint"         '""'  """^^ ''  '"  ^'^^^"^^  ^'^h  ^"^^h  multiplying  [ 

17.  The   product   of  two   nmnbers    divided    bv   one   ol  ^ 
them,  gives  the  other.  ^ 

18.  The   quotient   multiplied    by   the   divisor    gives    the- 
Jvidend;    the   dividend  divided  by   the  quotient  «iv?s  the- 


divisor. 


OK   THE   TWO    TERMS    OF    A    FRACTION. 


19.   ^very  fraction  is  composed  of  two  terms:  the  fr<t 
mentioned  is  called  the  numerator,  the  second  denominator 
ihe  numerator  indicates  how  many  equal  parts  of  the  unit 
are  contained  in  the  fraction  :  the  denominatk-  gives  the  name 
(»i  those  parts.  ^^ 

These  two  terms  of  a  fraction  are  assimilated  to  the  two 

crms  of  a  division.     The  numerator  represents  the  dividend  • 

the  denominator  the  divisor.  >'«t.iiu, 

20.  If  the  numerator  be  equal  to  the  denominator  the  frac- 
tion IS  equal  to  h  Jf  the  numerator  is  less  than  the  denoin- 
n.ator,  the  fraction  ,s  less  than  1.  If  the  numerator  is 
gi cater  than  the  denominator,  the  fraction  is  greater  than  1. 

21    Of  two  fractions  having  the   same  denominator,  the 
greatest  is  that  which  has  the   greatest  numerator.     Of  two  • 
fractions   having   the   same  numerator,  the  greatest  is  that 
which  has  the  smallest  denominator. 

22.  To  render  a  fraction  greater,  the  numerator  is  multi 
pi.e.  without  touching  the  denominator,  or  the  denominator  is 
divided  without  touching  the  numerator. 

lo  ronder  a  fraction  smaller,  the  numerator  I^  divided  mth. 
out  torching  the  denominator,  or  the  denominator  is  m„lt;. 
pnca  witijuul  louciiing  the  numerator. 

23.  Tlie  two  terms  of  a  fraction  being  multiplied  or  dlri 
d.vl  by  a  like  number,  its  value  remains  the  same. 


irt 


I    f 


S    f 


lii 


it 


1123 


APPEN  DIX. 


24.  Any  whole  numoer  may  always  be  put  indiferently 
vundei-  the  form  of  a  fraction :  ^he  only  thhig  is  to  give  it  the 

unit  for  denominator. 

25.  To  take  any  part  or  fraction  of  a  number,  is  to  mnlti- 
ply  it  by  that  fraction. 

Thus  to  take  the  |  of  12=12x|  =  ^^=8. 

Then  if  the  fraction  to  take  has  the  unit  for. numerator,  \vt 
have  only  to  divide  by  the  denominator.  Thus  to  take  the 
h-  h  01"  i,  &c.,  of  a  number,  is  to  divide  that  number  bv  2 
3,  or  4,  '  *^     ' 

20.  To  incrcfise  a  number  by  any  fraction  of  itself,  is  to 
multiply  it  by  a  new  fraction  whose  denominator  equals  the 
sum  of  the  two  given  terms,  and  whose  denominator  remains 
the  same, 

Ttus  to  increase^GO  by  ^^j=60Xi2  =  *  j§"=:85. 

ON   RATIOS    AND   PROPORTIONS. 

27.  The  ratio  of  two  numbers  is  the  quotient  of  the  first 
number  divided  by  the  second.    Thus  the  ratio  of  15  to  5  is  8. 

The  connection  of  two  equal  ratios  is  called  the  rjeometrlai, 
.  proporUoa.  Thus  15  :  5  : :  G  :  2  is  one  proportion,  as  the  ratio 
ot  15  to  5=3,  and  the  ratio  of  6  to  2=also  3. 

The  first  term  of  a  ratio  is  named  antecedent;  and  the 
second  consequent. 

Ihe  first  and  the  fourth  term  of  a  proportion  are  called 
ihQ  extremes  ;  the  second  and  third  are  called  \kii.  mediums. 

The  mediums  liiay  always  exchange  places  without  dis- 
turbing the  proportion. 

,  28.  The  product  of  the  extremes  alwavs  equals  that  of  the 
mediums. 

29.  We  determine  the  fourth  term  of  a  proportion  whnn 
'  unknown,  by  diuding  the  product  of  the  mediums  by  th  fir:^•t 
;  \erm. 


m  1 


ON    TflE    SQUARES     OF    NUMBERS    AND    THEIR    ROOTS. 

The  sqtiare  number  or  second  power  of  a  number  is  that 
number  multiplied  once  by  itself. 

Thu  ,sqi,<ne  m-H.  or  second  root  of  a  number  is  the  very 
number  which  has  been  raised  to  the  sauare. 


•  iSiiilfe^-jii'saai^  JiA 


APPENDIX. 


differently 
;ive  it  the 

5  to  multi- 


orator,  we 

take  the 

iber  by  ",1^ 

ts:clf',  is  to 
:quals  the 
»r  remauis 


129-^ 


*  the  first 
►  to  5  is  JJ. 

eometricui. 
3  the  ratio 

•  and  the 

ire  called 
'iiiums. 
hout   di* 

lat  of  the 

ion  whnn 
y-th  first 


)TS. 

r  is  that 
the   ver^ 


Tins  IS  the.  natural  series  of  squares  up  to  100 
Squares.     1  4  9  16  25  80  49  64  81   100 
Joots.        12  3    4    5     G     7     8     9     10 
(->bserve  that  the  successive  difference  befwpnn  f l.o 
*iways  exceeds  itself  by  two  un  ^^    "lT.n=   f  '^''*''?' 

30.  A  number  which  is  i  ot  a  nerfprf  cmm^^   •        ^^   ^ 

<^oot,  add  to  25  the  doubll  of  5lV^~i  i*     ^?    ^'''■'  ^  "'  *^^ 
25  +  11=36,  with  tlu^l^lfe'^"'^  ^'"^^  '"''^  ^^^^^ 

off^trlt-^I^ZrSot'r  ^  --^"^-'-t  i. 

Le   the  number  be  53,  of  which  the  v/=7+thc  rfm.ind.. 
-15  cut'offfh''  '^'  ^'.°'i'  '"'"^  ''''  do^.bIe  of  the    "ot  7  + 

64-15=49,  of  which  theXT      ^~"^  =  ^^'-^«"  ^^''^Ihave 

given  number,    ^'ampl .       '  '''^'  '^'  '^^'^^  ^^'^"^  ^he 

Sujiposc  tiio  number  is  86,  of  which  tho   /-o_Lfi 
>iov  5.     Ju  order  to  have  tl^  roo  -ft  f,!  '^~  i"^  uf  ''''^'","- 
rf,ot9_i_]7  nr]>^  ihl     ^'^.^oot=:8,  to   tho  doublc  of  tie 
^2.     Then l£lSf_nI  ^^f ''^•"/J^r  5,  you  will  have  17+5=. 

34.  Knowing  the  differonco  of  tw 


-xtractei 


M 


iu:nb.3rs  and   that   <jf 


m 

|:J^H 

s^^H 

;..■ 

H 

180 


APPENDIX. 


B*  11 


their  squares,  the  quotient  of  the  second  difference  divided  by 
the  first,  gives  the  sum  of  the  two  numbers,  which  enables  m 
to  determine  each  of  them  immediately. 

ON   THE    FACTORS   AND    SUBMULTIPLES    OF   A    NCMDER. 

35.  Every  number  has  at  least  two  factors;  it  may  I.it» 
more  If  it  have  but  two  factors,  it  is  necessarily  a  r ;•/;«« 
number,  (5  )  and  those  two  factors,  consequently,  are  that 
number  Itself  and  the  unit.  ,.  ^ 

36.  To  reproduce  by  multiplication  a  number  which  has 
niore  than  two  factors,  the  greater  factor  must  be  muliiplied 
by  the  lesser ;  ^ 

Or  the  factor  immediately  inferior  to  the  greater  by  the 
factor  immediately  superior  to  the  lesser;  so  on,  all  through, 
always  following  the  same  order.     Example  • 

Q  f 'f  Pq  "fo  ^^  5'i™^^''  ^^  ^'  ^'^'^^  ^^^  ^he  cifht  factors,  1.  2, 
^,  %  0, »,  1^,  and  24 ;  to  reproduce  that  number  we  shall  have  • 

24X1=24,  or  12x2=24,  or  8x3=24,  or  3X4  ^-24. 

In  this  example  the  quantity  of  the  factors  is  even:  if  it  be 
ooTrf  then  It  is  the  medium  factor  which,  multiplied  by  itself 
produces  the  number  in  question.     Example  •  ^  » 

o  !"EP?^^  *i!f  "7"^"^^  ^^  ^^'  ^^'^^  has  the  seven  factors,  1, 
*,  %  o,  10,  6Z,  04 ;  to  reproduce  that  number  we  shail  have- 

64X1=64,  or  34x2=64,  or  16x4=64,  or  8x8=64. 

^  From  this  article  we  shall  deduce  tiie  three  following  prin- 

37  The  greatest  factor  of  a  number  is  that  number  itself, 
and  Its  least  factor  is  the  unit. 

38.  The  greatest  factor  of  an  even  number  (that  number 
Itself  excepted)  is  always  the  half  of  that  number,  and  its 
least  factor  (the  unit  excepted)  is  always  2. 

39.  Every  number  which  has  an  uneven  quantity  of  fio- 
tors  IS  a  square  number  whose  root  is  the  mean  factor. 

40  Observe  that  every  number  has  always  a  submtdthU 
less  than  it  has  factors.  Indeed  a  number,  whatsoever  it  be. 
hgures  Itself  amongst  its  factors,  but  never  amongst  its  sub! 
multiples.  ° 

piication  of  two  of  its  sub  multiples,  the  unit  will  never  be 


diviclo4  ^y 
enables  ua 


MDER. 

may  I.^t? 
y  a  /r/?«« 
r,  are  that 

which  has 
multiplied 

T  by  tlie 
1  through, 

Jtors,  1,  2, 
hall  have  : 

iur24. 

;  if  it  he 
by  itself, 

'actors,  1, 
Kill  have: 

S=64. 

r'ing  prin- 

3er  itself! 

number 
.  and   its 

y  of  fio- 

r. 

or  it  be, 
its  sub- 

le  mui  ti- 
le ver  be 


appe:ndix 


131 


;l 


^1^^  tS:^^^;f  ^'  ^^^^"^  ^^-^  ^t  -not  in  any  way 
without  exception  ti  i^fuUi;!^^^^        ^'^  """^^^'  ^Ij 

•     ON    ODD    AND     EVEN   NUMBERS. 

Thl'of^tto'X'XrTl"    ir'"^  ^^'^"  even  number.^ 
an  ...  n.nbe.  andt^^j^^irbrisr.r  n.^^^^^^^  ^^ 

^^^^oV'^'t^^^^^^^^  J-  --numbers  is  an  e.en 

number.     That  between  «n.  ""x!"^""'  ''  ^^«^^  ^^^^  -^'^ 

i.s  an  odd  number  ''^  ""'^^'^  ^"^  ^"  «^^^  "umber 

Thlf  JMrLlb!^^  r—  n"-ber. 

--  number  and  an  ^ddZ^Ci.t  r ntnbJf "   ^^  ^" 
^,U,^  Every  oc^.„un.ber  has  only  o./cf  factors  and  submuL 

n^^^^'ZZ:^, "-''''  ''^  -^^^^  --Ption  of  the 

ON    PROGRESSIONS. 

4G.  Tne  arithmeticnl  vronre^^inn  1=  « 
cessively  increased  or  dimSed  bJ  .  ri.''''  ''^  *"""^  «"^ 
former  case  the  pro  JsZl  is  o.lL^    •  '  ^"^"''*^-     ^»  ^^^ 
latter  decreasing. ^  The  Sel  b  f     ^^^^rm^^ng,  and  in  the 
proportion,       "^  clitteience  between  the  terms  is  called 

The  geometrical  prooreisinn  \a  n  cr.  '        ly 
ly  multiplied  or  divided  bv.  l  if    '    ""^  ^^  *^''"'  successive. 
«i«etlmVogression  ssaid^toL."^"^^^^         ^"   the  former 

^^:!^:::?^;L:^;ir  :::i  ^J: -? -i^-  p-.-io. ; 

ii'^  terms,  mid  the  pyoporlion  or  ratio'    ''"   '"""'  '""  """  ''■'' 
47.  n,e  last  te™  of  an  ar.thme,:;.,  p..„g,,^i,„  ,.3  ^,„ 


ii 


1S2 


Al  PEXDIX. 


lili 


posed  of  the  first  tern,  more  us  many  times  the  ratio  a»  Ihert 
are  terma  before  it. 

^    Bxamplc-Su^pose  -2,  5,  8,  11,  14,  17,  of  which  the  ratio 
19  S.     Kemark  that  the  last  term  17,  which  has  Jive  terms  bo 
Kire^i^n  19  composed  of  the  first  term  2+ (the  proportion  3x 

^  48.  The  sum  of  the  terms  of  an  arithmetical  proffrcRsi(  n 
js  composed  of  the  sum  of  the  first  and  last  tefms.  muKipliod 
by  halt  the  number  of  terms.  ' 

^a«i^/..--Suppose  -5,  7,  9,  11,  13,  15,  of  which  the  half 
of  the  number  of  terms=:3.  Then  the  first  terms  5-ftho 
last  15=20  whichx3=60,  the  sum  of  the  six  terms. 

49.  In  all  arithmetical  progressions  the  sum  of  the  fir«»t 
and  last  terms  equals  that  of  the  second  and  last  but  one  or 
that  of  the  third  an(|  the  antepenultima,  and  soon  all  through, 
:5till  observnig  the  same  order. 

Example -Suppose^4,  7,  10,  13,  l^^  19,22,  25.  Remark- 
that  the  1st  and  the  8th  term =4 +25 =29 
that  the  2d  and  the  7th      "    =7+22=29 
that  the  3d  and  the  6th      "  =10+19=29 
that  the  4th  and  the  5th     "  =13+10=29. 
^    In  the  example  just  given  the  number  of  terms  is  even      U 
It  be  odd,  It  IS  then  the  double  of  the  middle  term,  which  equals 
the  sum  of  the  other  taken  two  by  two  as  above. 

^oram^^^.—Suppose  ^  1 1 ,  1 6,  2 1 ,  36,  3 1 .     Remark 
that  the  1st  and  the  5th  tcrm  =  ll+3l— 42- 
that  the  2ci  and  the  4lh      "     =16+26=42  • 
that  the  3d  doubled  =21+21=42.' 

50.  If  the  number  of  the  terms  of  an  arithmetical   pit., 
gicssion  be  odd,  the  middle  term  always  equals  the  sum  divi. 
ded  by  the  number  of  the  terms,  that  is  to  say,  that  if  tl  at 
number  be  3,  or  5,  or  7,  &c.,  the  middle  term =4  or  ^  or  1 
Kc,  of  the  sum  of  the  terms.  *       ?' 

^^a;w;>/^._Suppose  -^10,  20,  30,  40,  50,  of  which  the  sum 
=  150. 

Then  that   sum  divided  by  5,  the  number  of  the  terms 
—   5   =30. 
Hence,  as  we  see,  the  middle  term =30. 
If  the  number  of  the  terms  be  even,  the  two  mea^is  taken 


€ 

r 


J^|^|-f^^jrams '■- 


tio  a*  Iherft     v 

h  the  ratio 
'  term?  be 
ortion  3x 

rogros'ilc  n 
Tiultipliod 

h  the  half 
IS  5-f-thu 

IS. 

'  the  first 
ut  one,  or 
1  througli, 

.  Remark 


Al  PENDIX 


133 


even.     \{ 
eh  equals 

rk 


ical  pit), 
ium  divi. 
it  if  tl:al 
r  \  or  f 

the  suiw 
le  terina 

\s  takes 


i 


tS"'  '^'"^  ''^  ^""  ^'^'^^^  ^^  J-'^  the  number  of  the 

.umlTof  lfo?lT5  d'-.'^A'^'^'  2^'  3^'  «f  -h>ch  the 
terms=:io5-35      lln^Af  ^  ^^'  '^^  ^'^^^  ^he  number  ol 

first'mdtifiS  w'h^^'ati'riSrr     P^^^""^^'"'  ^^"'^^^  -^^ 
equal  to  th^  num/er  of  the  termsll     "  ^^''  '^  "^  ^^^^^ 

The,,,  oWve  tlrl^riS—  fel^Tx1.^=i^'- 

term  is  takon  from  the  product  3lH   ^      ■  ""f "  '  ""^  '''^' 
by  the  ratio,  lessened  by  a  uS  '''""""'^"  ''  ^'"'^"'' 

rJ'>t:t';fsrz:-o!o «/'« = ^,  ■■  '«2,  of  which  ,,,0 

=4b.      From  480   „".;«  the  fiTt  ™  o"  '^™   '"^^^ 
Now  tl,is  remainder  484  :  (thera.ro  t-xtZT^ll^!;  "^^ 
"r  the  terms.  "     ^^—   j   — -;4^,  sum 

p.y'^^r'i/V^o"  tf:if -'  P""f'-P'-  o„  progression  „p 

applicable  t:^„wrr;pfoS„?iri,o"', '" """'"  "■■™ 

change  the  words/,.,!^  Crdlf,  ilL^r"""^^  '^' 

ox   WV,SIBt„   NUMBERS   w,T„OUT  A   REMAINDER 

titule  o'^  t™nrS,:^S:ferthr '''r  '"  "  """ 
bcr  exactly  divisible  bv  another      Th  '^"'^^''  ^"^  """' 

n-de  without  any  remainder  ''"''  '""  "^'^-^  '-^'^'^^^  «'' 

oonsKiered  as  simple  unit  ™!w  it  ^ll^^^rt"^  '-^"■^^' 
rigl^atV^diSelfr  ^^'^^^^^^  ^^^^  ^^^  ^^  %-s  on  th. 

n;  ?"  nil  ""'"^''''  germinated  by  a  5  or  an  n 
i>>  0,  all  even  numbers  a 


lH 


12 


idy  divisible  by  3. 


i  I 


iU 


APPEXpiX. 


>  ■ill 


I'    Jili 


.if' 


•lit 


,       I 


000  Vc^'  '^^^'  ^^^^'  <^Cm  all  numbers  ending  with  0,  00. 

^J%}^\fSTJ''' ''''''""  """^bers  whereof  the  sum  of  :hi 
4rh  n h  ft..    !.       «'''f '  ^'^•'  ^'  ^^"^^  ^'^  the  sum  of  the  5>.] 
ik  19     II  ^^-'OrwhosedilFerenceis  11  or  a  multiple  of  IJ.' 
Ti    o-     1    ''^^''  numbers  already  divisible  bv  3  and  4 
liy  ^o,  all  <;i;en  or  «/jei;g;i  numbers  whereof  the  two  last 
hgures  on  the  right  are  divisible  by  25. 

PROPERTIES    AND    VARIOUS     EXPLANATIONS. 

4-fhi'  SI  ^^"^  TT^^l  'iT^'""  ^^^  S^-eatest  equal  (the  sum 
-f  the  d  flerence)  (fiyided  by  2,  and  the  smallest  equal  (tlS 
^wm-the  dfence)  divided  also  by  2;  whence  it  follow 
that  the  sum  mcreased  by  the  difTerenee  equals  .'e./crthe 
greater  number,  and  that,  diminished  by  the  difference  il 
equals  twice  the  smaller.  ^  -aicicace,  ii 

,.t^'Jf  ''*^?^  two  unequal  numbers,  wiihoui  altering  thetr 
sum  the  greater  ,s  dmanished  by  half  the  difference,  mid  the 
lesser  increased  by  the  other  half. 

50.  The  difference  between  two  numbers  cannot  be  superior 
LpiTse"  '''  '''-'''  """^^-'  ^-  ^^  -^  ^^-»  or 

67.  The  least  difference  possible  between  two  whole  num- 
bers, even,  or  between  two  whole  numbers,  od  I,  is  2 

58.  Knowing  how  many  times  A  is  greater  'than  B,  wo  at 
o^ice  perceive  how  many  times  B  is  smaller  than  A.  pre  e7wno 
the  same  numerator  to  the  given  fraction,  and  forming  «  new 
denomniator  from  the  sum  of  the  two  terms.     Examples- 
A  being  >  B  by  ^,  B  is  <A  by  4-. 
A  being  >  B  by  I,  B  is  <  A  by  /j^. 

stantly  determme  how  many  times  B  is  greater  than  A  pre- 
serving  the  same  numerator  to  th.  given  fraction,  and  flVrmTno 
a  new  denomniator  of  the  iriven  ,hmnu>u..,  ^„  a,  .  _.u.  ,  ",\"8 
numeratcn- is  subtracted.     Axamples  :  ' •— u  uio 


figures  on 

hose  ligurc:! 

vith  0,  00. 

sim»  of  :1m 
of  ilie  JM, 
tipleof  li. 
and  4. 

e  two  last 


APPEXDIX. 


(the  sum 
equal  (the 
it  follows 

twice  the 
fcrence,  it 

nng   tketr 
e,  and  the 

e  superior 
equal  or 


lole 


II  urn- 


B,  wo  at 
reserving 
ng  &  new 
npks  : 


3,  we  in- 

•  A,  pre- 
formino 


A  benig  <  13  by  |-,  13  is  >  A  bv  t' 


i&c 


of  the  givol  fractiot    iw«>  '°™"^"*  "'«  '»•"  '«■•- 

Aboing|ofB,  (i=i„,^A.' 
Abcingjof  iJ,U=|of  A 

co,fmi„f;h:ii.iit:„rw'„  ir  '•■"'  r"^  "^  •- ""-"'-«. 

number  to  the  o.^.^^Z^'Ch.Z^''.  P"''""'""  "^  -- 

«"«  its  den„n,i„ator.    S  seco,  d  .w'™'''";,  "'^  ""=   ''■•«'' 
proportion  of  the  lesser  m,nbert„  .h  *'"  ""^'^^^  ">'■• 

it  will  expres.s  the  nroporZn  of  ,  f         S'™"'"""'  """^  "»"^''i 

i'-^am^k.-T.^vrtT^'^Zl  13 T'"f  'V"?  'r"^- 
12  and  the  difference  2-  the  ™!„  „     .    '  °^*l'"'''  thesu,n  = 

the  other.  Then  6  in 'a  ,Ltbn  rC'''^  ""T'^"'  «  "'"'^ 
second  fraction  such  as  it  i,  we  «1>  S,  1=''  "'"'  '^'"«"'g  «" 
t'.at  the  lesser  nn,„b:r=V:fThV",,t-''  *•  "»'  -  to'  sav 

the  s„„,=95  ™f  Ihe  d  Le,T-'°"  '  "  ^'"'^  ^=-  "^  -^^ 
«,„.  o.:.__     .    .        ^  aincrei.ce  2.        -ie  ,„e  contains,  there 


ore  3  tunes  J  the  other.     Then 


t  differ  bv  9  fh*.  ^.-fli;  ">  by  0  or  a  multiple  of  9    W 

itditrerby^^fotb;?"^^^^^^  isV'  ,f 

the  two  figures  is  2  3    W      r^     \^^''  ^^^<^'fferencebetM.>  o 

fta    Tk«         II        ,    "'"erence  of  the  figures  H  nnA  i     t 
tW.  The  smaller  the  difference  -^  hr.u.         .         ^~^- 


No.  1. 
Sum  a 


3x5=15      Sum  9. 
,4x4=l(> 


^X  /  =  14 
3X6=18 
4X5=20 


« 


'  fu 


I'if 


186 


APPENDIX. 


1st.  The  one  of  the  two  numbers  which,  laultiplied  one  bv  tho 
other,  g.vc  the  smallest  product,  is  alwkys  tht  un^C^hethe 
thi;  sum  be  ocW  or  even;  '  ''"*'"'*'^ 

'o  ,   f.'f,  ^^'°  S'-eator  product  is  equal  to  half  the  sum; 

od.  if  the  sum  he  odd,  No.  2,  the  two  numbers  M'hich  give 
the  g,  eater  product  differ  between  themselves  by  the  unit. 

04.  There  are  some  quantities  which,  according  to  the 
lutuie  ot  the  question,  can  only  be  fractionary.  Thus  fov 
n.stance,  when,  m  speaking  of  workmen,  birds,  eqqs  &c '  wg 
mentioix  the  /.«//;  thirds^  Quarters,  &c.,  it  is  n^cefsarit  kp- 
po  ed  that  those  quantities  are  exactly  divisible  by  2, 3,4,  &c. 

flf!  nf  /^  k'!^  Tl  ^''"'''^'>''  ^^^  ""»^^«r  whereby  i  quan- 
tity of  his  kind  becomes  divisible,  if  increased  by  one  of  it. 
parts  add  the  two  terms  of  the  fraction  which  expresses  that 
part,  the  sum  will  be  the  answer. 

\:ollT'  ^'''  f '""Ple,  if  1  increase  by  \  the  contents  of  a  bas- 
kct  of  eggs,  I  conclude  that  those  contents,  at?  first  exactly  di- 
visible  by  7,  is  now  divisible  by^+5=A   12 

If,  on  the  contrary,  the  quantit/of  one'of  its  parts  be  di- 
«  .imshed,  you  will  determine  the  number  by  which  it  becomes 
divisible,  taking  the  numerator  from  the  denominator.  The 
relTianider  will  be  the  answer. 

I    '^^^".^'/^'•/^^a'^Pje,  if  I  diminish  by  f  the  birds  of  an  aviarv 
I  conclude   hat  their  number,  at  first  exactlv  divisible  by  7 Ts- 
now  (divisible  by  7— 5  =  A.  2.  "  ":/*,^^ 

In  fv/.^l  'f  ?K  ^P'^'f «"  ^'hich  often  occurs  in  my  solutions. 
In  Oder  that  the  reader  may  perfectly  understand  it,  J  am    ' 
flbout  te  give  here  an  explanatory  example. 

Let  us  suppose  that  the  question  is  to  divide  |'r4-4^  i„fn 
tvjo  parts,  one  of  which  =  the  f  of  the  other.  Make  the  su.u 
o(  the  two  terms  of  the  given  fraction,  you  will  have  S-f  5  ^ 
N  which  indicates  that  the  lesser  part  ought  to  have  the  2  -»i 
the  number  (a:+4)  and  the  greater  the  |!  * 

Opera*   n  ;      (a:+4)  f  =:(2f+i^  the  lesser  number. 


m^ 


t>+4}|  = 


8 


the  greater  number. 


1th  thcvfol 


APPENDIX. 


1  PT 


one  by  tha 
I'fc,  whether 

1  numbers 
sum; 

which  givo 
he  unit. 

ng   to   (he 
Thus,  fov 

S,    &C.,  AVG 

iarily  sup- 
i,  3,  4,  &;c. 
)y  a  quan- 
one  of  its 
esses  that 

I  of  a  bas- 
ixactly  di- 
rts be  di- 
;  becomes 
tor.     The 

Ml  aviary, 
e  by 


7 ,  IS  • 


solutions, 
it,  1  am 

f  4)  into 
the  sum 

s  3-1-5^ 
the  I  «)i 


aer. 


MISCELLANEOUS  PJiOB]  EMS. 

lien"-  o?S!ft"erS"?„rr  "?'  *«  ^"'"  <"■''-  'i-. 
•nay  be  10.  ''"«  """  'V  7  and  tho  other  by  3 

07.  Divide  *lfrft  1   .  ,  -^'(s.  28  and  IS. 

t-nally  to  theif ag^s  :'b  W  t"?  TT  ^'  ^'  ^'  P^P- 

ofA,whoisbutL]fofC'rWh^   ;[^?'^^^^^  ^^^"  ^'^'^t 

J       A  *^  *^*^  ^^^''e  of  each  ?-' 

What' is  the  capital V^*'        '"''"'  '^  ^^^  ^'"«"»t  of  $8208. 
69.  A  capitalist  placed  the  *  of  V      .    ,    "^''''  ^^^^^' 

;...o,-e  arte,,  .hid.  \Z2^,^  mo'"fr''  "'"  '""  '"  *''» 
hi-.st?  "-"  ''-'"•     Wow  many  were  in  ii 

'2-  Says  A  to  B,  Give  ,„c  *10(.       i  ,    "'"'•  *^^- 

«l.at  thou  hast.     Ill m^U'ad  «lTV,.f 'i-nlf  ™/'""'''« 
'3.  Find  two  „u,„he.s  whose  dr  *''"''""'^  ®™"- 

-y  be  to  one  a^^o^J' :^-J:t:--<'^;;,  r  ''"""''• 

7r,     A       J  r»  ^^"'^'^  are  the  numbers?     Ans    ^  m^A  a 

''    would  ha™  .;s'm™h  kJ  A Imd'clrr^",^--^'^"^ 


(ais  lii 


IS  each  ] 


--■^^.mnutcr^t 


■is  I 


-'J 


133 


APPENDIX. 


76.  What  is  that  number  whose  seventh  part  multiplied  bv 
Its  eighth  and  the  product  divided  by  3  would  give  298|  for 

It.  What  are  two  numbers  whose  product  is  750  and 
whose  quotient  is  3^  ?  Aus.  50  and  15. 

78.  A  person  being  questioned  about  his  age,  replied  •  My 
mother  was  twenty  years  of  age  at  my  birth,  and  the  nun.boV 
of  her  years  multiplied  by  mine  exceed  by  2500  years  hgr 
ige  and  mine  united.     What  is  his  age  ?  Jus.  42. 

.eJ^'i-^ iffl^T"^  ^,^"S^^  ^^"^^  furniture  and  sold  it  shortly 
after  for  |1 44;  by  which  he  gained  as  much  per  cent,  as  k 
CA>st.     Kequired  the  first  cost.  Ans.  $80. 

80.  Determine  two  numbers  whose  sum  shall  be  41  and 
the  sum  of  whose  squares  shall  be  901.       Ans.  15  and  26. 

81.  The  difference  between  two  numbers  is  8,  and  the  sum 
ot  their  squares  544.     What  are  the  numbers  ? 

8x5.  Ihe  product  of  two  numbers  is  255.  and  the  sum  of 
their  squares  514.  What  are  the  numbers?    Ans.  15  and  17. 

83.  Divide  the  number  16  into  two  such  parts,  that  if  to 
their  product  the  sum  of  their  squares  be  added,  the  result 
will  be  208.  Ans.  4  and  12. 

^■^V^^*  l^o^i'^  "'""^^''  '^^'''^  ^^^^^  to  its  square  root  the 
sum  shall  be  1332?  W  1296. 

85.  What  is  the  number  that  exceeds  its  square  root  by 
*'  Ans.5Ql. 

?^L  ^i"i  *^°  numbers,  such  that  their  sum,  their  product, 
and  the  difference  of  their  squares,  may  be  equal  ? 

Ans  .^tVi>    l±-/5 
2      '       2~* 

87.  Find  two  numbers,  whose  difference  multiplied  by  the 
difterencc  of  their  squares  gives  for  product  160,  and  4ose 
sum  multiplied  by  the  sum  of  their  squares  gives  580  for  pro 

fit   .V,  Ans.7md2. 

88.  VN  hat  is  the  ratio  of  a  progression  by  difference  of  22 
terriM,  the  first  of  which  is  1  and  the  last  15?  Ans  ^ 

89.  J  here  is  «  nmviKni.  ^e  t.^.-^  £. _--        «    .i  _.  .^ 

vide  It  by  the  sum  of  its  figures,  then  inverting  the  numbci 


rjultiplied  bjj 
ive  298 1  for 
Ans.  224. 

is  750  and 
•0  and  15. 

•eplied :  M/ 
tho  number 

0  years  hsr 
Ans,  42. 

Id  it  shortly 
cent,  as  it 
dns.  $80. 

be  41,  and 
3  and  26. 

.nd  the  sun) 

1  and  20.    " 
the  sum  (»f 

15  and  17. 
,  that  if  to 
,  the  result 
t  and  12. 

ire  root  the 
IS.  1296. 

•e  root  by 
ns.  56^. 

sir  product, 

i  +  v^s 

2~* 

ied  by  the 
and  whose 
^0  for  pro 
r  and  3. 

snce  of  22 
Ans.  |. 

ii  yuu  di. 
i  numbci 


APPEXDIX. 


139 


lot,  I  answeirSMhe  »Tf  ?.?'  t«f  "-''^  P"'  "  »  ■-"' 
of  «n  egg  exceeded  the  f'of  the  wS  h^T '"'''''  ^•■»-  ">«  J 
all  the  eggs  employed  "''"  ''>'  "■«  ^<i"'"-e  roof  of 

n.e'fh.tergTTdf  S  Te  ""'  ■""?'"'»"•  •""  '^  «'vo'L 
instead   of  £5°  that   I   lost  fr""^'"'^'"'  »/ "vy  n,„„ev,  an.l 

many  pounds  have  I  now^'         ™  ""*"'  •^"'  §'""•      ""w 

»"Mnrtr:to';?pil%^H„d.^^ 

dren  have*!?  '^'^  '">^  ^^""'y-     How  many  ehil- 

96.  Two  numbers  are  sunh  fKof  ♦  •  7     ,     ,       ^'^"''''  ^^• 
more  than  the  greater    and  if  ft     *"^^'  ^t"  ^^'^  '«  3   units 
i\  and  the  U^s^'tsT  the  fori    h  ^''"'''  ^  augmented  its 
What  are  they  ?      *'  ™^'    ^^^"'"^'^  <^ouble  the  latter 

97.  A  brother  said  to  his  sister  •  f  m^'"'  ^^  ""^  ^^• 
founds  to  make  £30-  eive  niHl'      T"^^  ''^''^  *  «^  vour 
t^e  sister,  |  of  yours'to^ havellt  will   v""''  """^-  ^^^^'''^^ 
How  many  pounds  had  each%       '          /^"  gT."  '"«  ^^em  ? 

yS.  By  means  of  „   i         "     .  ''•  "^^  ""<^  -^^0. 

Michael  L  spend  $2  a  dTan^^^^^^^^  ^"'"^"'^'^'^  ^'^   '— ^ 
legacy.     Whit  was  it?      ^    ^  '^^  "^  ^^«'-'>'  ^he  |^  of  tho 

99.  A  fruiterer  )mn„Kf  ^''*-  *^00. 


J  n 


14U 


APPENi  IX 


jvould  have  been  but  10  shillings  less  than  half  the  first  cost 
Itoquu-ed  the  first  cost  and  selling  price. 

Ans.  100  and  120  shillings. 

101.  Determine  two  numbers  whose  difference  shall  b« 
equal  to  one  ol  them,  and  whose  product  shall  be  18  more 
than  triple  their  sum.  Ans.  12  and  6. 

102.  Of  three  numbers  the  mean  is  ^  greater  than  the  less, 
jnd  the  turmc-r  is  i  less  .han  the  greater  ;  now  if  each  was  re- 
rtuced  ^,  tiieir  sum  would  be  reduced  19  units.  What  are 
the  three  numbers  ?  Ans.  24,  18,  and  15. 

103.  Three    times  a  number,  less  20,  is  as   much  above 

fr^V   wr^'-"T^^'"^f  ^^'  ^°"'^^  part,  plus  2,  is  undei 
Its  half.     What  is  the  number  ?  Ans.  24. 

104.  Louisa  buys  2^  lbs.  of  sugar,  at  6d.  per  lb.,  andgives 
in  payment  a  piece  of  silver  such  that  the  square  of  the  Siece 
returned  exceeds  trie  triple  of  the  expense  by  a  sum  equal  to 
the  return.     Required  the  value  of  the  piece  given  by  Louisa. 

Ans.  "is.  8d. 

105.  By  what  number  should  3  be  multiplied  in  order  that 
the  f  J  of  the  product  may  be  equal  to  the  sum  of  the  two 
^^^^^'''^ '  Ans.  By  12. 

106.  When  the  granddaughter  was  born,  the  grandfather 
was  3^  times  the  granddaughter's  present  age,  and  10  years 
af^er,  the  latter  was  but  i  of  the  grandfathers  aforesaid^a«e. 
Kequired  their  respective  ages.  Ans.  90  and  20?^ 

107.  Determine  three  numbers,  of  which  the  greater,  equal 
U)  the  sum  of  the  two  others,  is  also  equal  to  the  I  of  their 
product,  and  of  which  the  less  is  but  the  I  of  the  other  two 
^^''^'''  aI.  24,  IS,  and  G. 

1  ^^?"..^"  yncle  claims  the  /^  of  an  inheritance,  the  nephev* 
I  and  the  niece  the  remainder.  The  product  of  the  parts  of 
the  two  latter  IS  13  millions  less  than  the  square  of  the  m- 
clespart.     What  was  the  inheritance?  ^ns.  $12  000. 

.,\^^:.y''^'^^'^<'J'»^^^^^^^'^,yvhose    sum    is  equal  to4  time?* 
their  difference,  and  the  greater  of  which,  plus  the  difference 
?Kceeds  the  less  by  48  units.     What  are  the  numbers  ? 

Ans.  ()0  and  3t>. 
My  wafch  is  very  methodical  in  its  time.     Were  (  to. 


iiU. 


ie  first  costii 

shillings, 
ce  shall   be 
>e  18  more 
12  and  6. 

lan  the  less, 
iach  was  re- 
What  are 
'<,  and  15. 

luch  above 
2,  is  undei 
Ans.  24. 

».,  andgives 
•f  the  piece 
m  equal  to 
by  Ix)ui8a. 
f.  Is.  8d. 

order  that 
>f  the  two 
.  By  12. 

grandfather 
1  10  years 
•esaid  age. 
and  20. 

ater,  equal 
J  of  their 
other  two 
,  and  G. 

le  nephew 
e  pai-ts  of 
of  the  lUi- 
12,000. 

to 4  time?* 
lirteronce, 

TS? 

md  3tj. 
Wan-  f  tf> 


APPENDIX. 


141 


1 1 1  -p^,,,  ,  ^^^'  i  minute  fast  or  slow 

od'r\ZZrCti7;.'  Z:':*"'  T  ^  <'^*-^  brother-  Tea,. 
J  of  the  sister's  ^eatiesHL/nT  -T"  «•■«  brother  will  be 
ofeaoh?  ^^O's  less  than  the  sister.     What  is  the  age 

112  HnH  I  ^     u,  ■  ®'"'^'"'  '8;  lirother,  U 

<  -M  Hve  tripled  ^.r^^bral' tt  |.  J^'J'  ^^ 

^■^rz\r'tt^-^ttfX-:i.ei 

114    I  k„  .  ^^"•*'-  -A^t  72  and  at  40 

-^-  .'ow^^;israri::;rz'^«  "-^ '?— 

net,  just  as  you  please  nf  ,ZT  [I    *"«'■■""'•  the  prod- 
my  sister's  ai;d  Zt^:/'^^ .^;::'JX  "^"^     '''--«■"' 

115.  Subtract /oT.';tot3''^  '--■  2  years  each. 
««e,you  shall   hlem^Lthit,''-''^''-T'J''''<"y^^tor', 
years  will  have  lessen^lt  " Wh^^lhTf ,f„f ^ h  f  '"^'^ 

onVo^r  teoiro^tf  s  but  »^— '^^  • 

.u.  is  but  half  ,„,  ,J,  ^Z:  I^l^tstelt'' 

mayi4"„^airjTtl;:rat'  '"1 1  ^''™-  o^'"  I-' 

lt«  i,  may  exceed'the  lessftv  I'  "  ®?'"''  *"""'f-ed 

US   Tk         J  ^«».  20  and  4. 

.«m,  and  thr,ttie"„t  Ts  "  Wh^  ^"^'."'^  '^^^^  «--  '''eir 
4     i"«"r  IS  rf.      vv  hat  are  the  numbers  ? 

119.  Square  I  plus  1  of  thn  .     ^''''  ^^  ^''"^  "^ 

qimrter  of  a  n«n?„f,   1.  ^.^*^^.3r««»•«  tti.'it   I  am  .hort  of. 

r*  ?  ""  " "'  '^""  •> ""  ^"'  P'-oduce  my  age.     What  is 


120    A  gamester  lo8»  at  the  first 


Am.  16. 


game  the  srjujire  of  ^  o| 


142 


APPENDIX. 


the  dollars  he  ha4  about  him;  but  at  the  second  round  he 
quintuples  his  remainder,  and  withdraws  neither  gainer  nor 
loser.     How  many  dollars  had  he  ?  Ans.  80. 

121.  I  am  going  to  add  5  new  shelves  to  my  library,  each 
of  which  will  hold  20  vol'umes  more  than  the  ten  already  ex- 
isting and  so  I  shall  have  1000  volumes  in  all.  How  manv 
^'«^^^"«^^  ^...600.  ^ 

122  Multiply  half  the  father's  age  by  half  the  son's  age  and 
you  will  have  the  square  of  the  son's  age;  this  square  is 
equal  to  double  the  sum  of  their  ages.     How  old  is  each  ? 

Ans.  40  and  10. 

123.  The  breadth  of  my  room  is  but  the  f  of  its  length  — 
As  broad  as  it  is  lo-jg,  it  would  contain  144  square  feet  more. 
Kequired  its  dimensions.  ^ns.  24  by  18 

124.  The  difference  between  the  |  of  my  age,  less  5,  and 
Its  g,  plus  3,  IS  the  square  root  of  my  age.     What  is  it? 

Ans.  3(5  years. 

125.  The  product  of  two  numbers  is  220.  If  from  tho 
greater  you  subtract  the  difference,  their  product  will  lessen 
yy  units,     i^md  the  two  numbers  by  one  unknown  term. 

Ans.  20  and  11. 

126.  What  is  the  number  of  your  house?  Tlie  sum  of  its 
digits,  considered  as  units,  is  equal  to  ^  of  the  number.  Find 
*^  Ans.  54. 

127.  The  square  of  the  difference  of  two  numbers  is  equal 
to  Its  sum,  and  I  of  the  former  is  equal  to  |  of  the  latter. 
What  are  the  two  numbers  ?  Jns.  10  and  6. 

128.  An  officer  gave  the  following  indicatjon  of  the  number 
of  his  regiment :    One  of  its  factors  is  to  the  other  •  •  1  •  5 
and  their  sum  is  to  their  product  ::  6  :  25.     What  was  thr 

luniiber?  ^       ,o- 

Ans.  12a. 

129.  My  age  is  composed  of  two  figures,  and  read  back- 
wards It  makes  me  I  older.     What  is  it?  Ans.  45. 

130.  The  product  of  two  numbers  is  ^  more  than  their 
sum,  and  is  equal  to  triple  their  difference.     What  are  they  ? 

Ans.  2  and  6. 

131.  When  the  brother's  age  was  the  snu.irf^  nf  fh^  =-!=f.-.T-.»= 
/he  was  |  of  the  brother's  present  age,  and  8  years  hencrthe 


round    he 
gainer  nor 
Ans.  80. 

irary,  each 
il ready  ex- 
low  many 
ns.  600. 

n's  age  and 
5  square  is 
i  each  ? 
and  10. 

?  length. — 
feet  more. 
t  by  18. 

ess  5,  and 

is  it? 
>  years. 

'  from  tho 
nil  lessen 
term, 
and  11. 

3um  of  it8 
>er.  Find 
ns.  54. 
's  is  equal 
he  latter. 
'  and  6. 

e  number 
?r  : :  1  :  5, 
It  was  thr 
*5.  125. 

jad  back 
ns.  45. 

h.in  their 
^re  they  ? 
and  6. 

.^  „:_i. »_ 

hence  thr 


APPENDIX. 


143 


«  their  ages  will  be  augmented  it.  I.     What  is  the  age 

and  at  the  sam?  perioj  th^eTe  of  ^    ''''"  ^'  ^"'  '^^  t  ^^^''■^ 
i  of  the  three  agL  united'^    f  hi  te  the'^^S^  '""^''^^^'^ 

134.  Divide  a  baskpf  .f  ^'''*  ^^'  ^^'  «"<^  1^2. 

the  part  of  the'  o^t  m  ;2^thaT^f  .t"^  ^'■^^^'•^'  ^  ^^^^^ 
aiid  that  of  the  second  to  the  parf    f   h     '  '''^"'^  =  =  *  ^  ^  5 

greatest   ^ubmultS  ?    i     d2'"''h  *'"*!  divided  hy    t. 
pounds  d'd  he  gain  ?'  ^^'I'shed  £12.       How  manj 

number.     What  is  it  f  °"^"  "'  ""^"itude,  i,  f  „f  the 

137    J  gQ]^  ,    ^  -4wa-,  JiH. 

in  anoiher.     Triple  tto  r'mri^,dl"*^1  ''"  ""'  house,  and  25 
tlie  primitive  ccftents.     What  h^'  it"     ^""  ^'""'  '•''P'-"i''oe 

">y'?^t;&,;itS;re?ttsv''^--^^ 
r^i^eS  etir  ^■"'  —  "z:i  mZ..i,- 

I.IQ    rr  I  K„^      -11  ^'**'  1^  and  8. 

Jave  bee^lettr^*  CltohaTlf  •  '1  "'^  -"'" 
a«d  I  pay  for  it?  "uuuie  wiiat  it  cost  me.     What 

140.  One  of  the  fapfnr«  „p  ,  •  ,  ^''*-  ^^'- 

■■"" "  -  "■■  ■"--  -att  !;ttt 


IS 


144 


APPENDIX. 


pres,  ut  time,  plu^  its  root,  is  half  greater  than  less  its  root 
What  o'clock  was  it  ?  4^^  g 

142.  The  sum  of  the  four  terms  of  an  arithmetical  pro. 
gression  is  44  ;  tliat  of  the  two  first,  18.  What  are  the  four 
^^""'^-  ^1/^^.8,10,12,14. 

143  There  is  4  difference  between  two  numbers,  and  theii 
sum  is  less  tiian  their  product.     Required  the  two  numbers. 

Ans.  2  and  6. 
144.  The  difference  of  two  numbers  equals  4  of  the  jjreater 
and  represents  the  square  of  the  less.     What   are  the  num' 
^^^^-  ^ln«.  30and5. 

145   There  are  two  unequal  numbers;  the  less  is  equal  to 
I  ot  their  sum,  and  their  sum  equals  ^j  of  their  pr  )duct. 

Ans.  6  and  4. 

ini^^'  I^u  '"??^"^  ^'^^  numbers,  plus  their  difference,  makes 
100,  and  their  difference  joined  to  their  quotient  equals  45 
What  are  the  numbers?  Atis.  50  and  10.    * 

147  I  received  this  morning  a  basket  of  peaches:  I  laid 
i  by  for  myself,  and  the  remainder,  a  prime  number,  1  divided 
amongst  my  children  in  equal  parts.  How  many  children 
^^^^^^  ;in..  11. 

148.  The  I  of  the  numerator  equals  the  f  of  the  denomi- 
nator, and  the  sum  of  the  two  terms  is  11  more  than  the  pro. 
duct  of  ^  of  the  denominator  by  ^  of  the  numerator.  What 
IS  the  fraction?  j„„   s 

149.  1  bought  a  horse  yesterday  and  sold  him  at  a  profit 
equal  to  the  f ,  less  £11,  of  my  outlay,  bv  which  1  gained  20 
per  cent.     Required  the  first  cost  and  selling  price. 

Ans.  i>20  and  £24. 

150.  The  four  terms  of  a  proportion  make,  together,  100. 
Ihe  first  i«  equal  to  the  third  and  the  ratio  is  4  What  are 
the  terms?  Ans.  40  :  10::40:  lo! 

151.  The  dividend  is  equa'  10  the  square  of  the  divisor  and 
their  sum  added  to  their  qu  nent,  is  equal  to  840.  What  is 
the  dividend  ?  what  is  the  divisor  ?  Ans.  784  ar.  d  28. 

152.  One  square  is  quadruple  another,  and  their  sum  added 

to    the    sum    of  the    t.wn     rm^ta    \a    f^QA         \,\7l.„*.     .1 

—  -    ...  .,._.._.,       TT  iK-tL    arc    tne    twc 

^"^^^^«-  ^-Im'.  400  and  100. 


APPENDIX. 


3S  its  root 
Ans.  5. 

etical  pro- 
re  the  four 
,  12,  14. 

,  and  theii 
umbers. 
2  and  6. 

be  greater, 
the  nura 

>  and  5. 

5  equal  to 
)duct. 

>  and  4. 
ice,  makes 
squals  45, 
and  10. 

!s:  I  laid 
,  1  divided 
children 
ns.  11. 

denomi- 
1  the  pro- 
'.     What 

A.ns.  |. 

it  a  profit 
gained  20 

di;24. 

her,  100. 
•Vhat  are 
0:10. 

isor,  and 
What  is 
r.d  28. 

m  added 
the  two 
i  100. 


145 


153.  A  milkmaid  sells  hens'  eggs  and  ducks'  oa^.  tu  • 
mean  price  is  16  cents  the  dozen  Now  a  a  ^f '.  .^^''"" 
are  worth  10  dozen  of  the  former  pin  •  1T"  ^^  '^'  ^^"*^^ 
dozen.  "'^'''     Required  the  price  of  each 

.r.    „ru  '^'**-  J 2  and  20  cents. 

io4    What  cost  these  sl^  pounds  of  sucrar  ?     rr  .k 
»^orth  3  cents  per  lb  more  mv  nnM       ^    ,1    .     ^^^^  ^^""o 
more.     Whatliditcorre'rT/.       '^  would   have   been  I 

l^r      m,       .  ^  ••  ^ns.    iDctS. 

they «  ^       *"  "^  '""  proportional.     What  are 

two  numbers?  -^  '^^     What  are  the 

1K7     A  ,  ^w*.  6  and  6. 

.  .xth\\etSe'rXbr4:taVt  roof  "r^^/T 

square?  'i  "mes  its  root.      Required   the 

•     158.  An  annuity  placed  at  iqi  r..  .  ^"'''  *^^' 

duces  monthly  a  lenteauJfn-J  ^      ''^"*'  P^*"  ^""'""'  P'"^ 
annuity  ?         ^  ^^"^^  ^^  ^^"  ^^^^''^  '^ot.     What  is  the 

i^jf.   Ihe  product  of  two  numbers  is  120      A,l^  i  .  l 

and  their  product  shall  be  150.     wLt  a '^he dumber;  ?'"^ 

160.  Two  numbers  are  equal      If  o  k.    fj'l'  ^  ^"^  ^• 

product  will  increase  51      What  are  Th.  !    ^^^  '?  ''^"^'  '^^'^ 

«»  »'i.      w  nat  are  the  two  numbers  ? 

Iflo   Tk« «      1-        -  '^'**'  4  and  12 

ho  less  b,  2.     m  Jr  th'/thtettr  ""^  <i-''™'  »"' 

164.  Th«  «n„„..«  „,  .u.  ..  .  '^"'-  H  4,  and  2l'. 

nominatorrandThes'.lm'nfZ?'''''?'''''.'""™  "■»"  *«  dfr 
their  difrer'enee  '"wh,™  r/the  fra^tioT""  "  '  "'""'  '"l"'"  T'-^ 


146 


APPKiVDIX. 


1«5.  The  two  torms  of  a  division  mm  are  the  san  e  as  the 
two  te:  ,ns  of  another,  but  in  an  inverted  order,  ^e  sum  ot' 
the  fimr  tenns  is  80,  and  that  of  their  Quotients  2i  What 
are  the  terms  of  these  division  sums?    '    Am.  24  and  hi ' 

«.n^^t"  i!'^'5  ^]'.^  d'^'^^nfl  is  equal  to  the  square  of  the  divi- 
sor ;  halt  the  divisor  equals  the  square  of  the  quotient.  Re- 
quired  the  <lividend  and  divisor.  Am.  128  .and  8 

tn  lTrl'P^7u  """""^r  ^?  V  ®  ''  ^-  ^"^^'"^^^  5  f'-'^'"  the  first 
to  add  to  the  -econd,  and  they  will  be  : :  7  :  6.  What  are  thp 
two  numbers?  ^„„  ^n     ^  «k 

-^"«.  40  and  25. 

«nn  ^®Vt"''^'  'V^^/f '^««  fro"^  A  to  B  ?  said  an  inquisitive  per 
son  He  received  for  answer  that  their  numbor  had  but  two 
factors,-  whose  sum  was  20.     What  is  that  number? 

TT,  Ans.  19. 

A^i  ,  u  P^'^'^'J^^^Pf  the  two  terms  of  a  fracti..n  is  120  — 
Add  1  to  the  numerator,  subtract  1  from  the  denominator  their 
quotient  will  be  1.     What  is  the  fraction  ?  Ans!  '{^ 

17.0.  The  failure  of  an  insolvent  debtor  took  away  the  !  of 
the  capital  that  .had  placed  in  his  hands.  The  interest  of' the 
remamder  placed  at  5  per  cent,  is  equal  to  the  square  root  of 
the  hrst  capital.     What  was  it?  ^n.%10,000. 

,..^^^-i/^  ^?"  ^""^  '"^^  ^^"^  ""^'^^  ^^^^  "^y  ag«  backwards,  and 
you  will  make  me  the  J  younger.      What  was  the  person's 

^^  •        ,  Ans.  81. 

,x,^lu''  F''f  "^.^  t^'ee  numbers  in  arithmetical  progression 
the  third  of  which  shall  be  equal  to  the  square  of  the  firTt' 
and  the  second  triple  the  ratio.  Ans.  2  3  4 

*.  '"'^n  J  i?T"u^^  ,^'  ,""'^^^  ^"  hour,  said  a  pedestrian  :  had  I 
travelled  7^,  I  should  have  arrived  8  hours  sooner.     Required 

the  length  of  the  journey.  Ans.  240  miles. 

174.  Jf  you  double  the  denominator  of  a  fraction,  the  sum 

l'ln'''''-irTT"Jf  22^  and  if  you  triple  its  numerator, 
the  sum  will  only  be  21.     Determine  the  fraction,    ^/j,.  |. 

175.  Increase  the  contents  of  my  purse  £3,  and  it  will  be- 
come,  a  perfect  square.  If,  on  the  contrary,  you  lessen  it  £3 
di^pin  w!T  ^"^^^  ^'  *^®  aforesaid  square.     What  are 

176.  A  lady  questioned  about  her  age,  answered  :  Increase 


APPENDIX. 


^an  e  as. the 
rhe  sum  of 
II     What 
and  1 «. 

)f  the  divi- 
tient.  Re- 
8  and  8. 

Ti  the  first 
Kat  are  thp 
and  25. 

isitivo  per 
d  but  two 

ins.  19. 

is  120— 
lator.  their 
Ins.  f^. 

i  the  I  of 
est  of  the 
i"e  root  of 
10,000. 

ards,  and 
person's 
ns.  81. 

)gression, 
the  first, 
i,  3,  4. 

n  :  had  I 
Required 

miles, 
the  sum 
merator, 
[ns.  I 

will  be- 
en it  £3, 

Vhat  are 
s,  £6, 

Increas* 


w 


it  the  f,  and  in  that  state  lessen  it  the  *  and   vn,,  .h.u  u 
made  mo  25  years  younger.     What  wat  the  la'd^s  age  ?  ''^^ 

resent*  my^tT  '  ^  "butlf'^  ^-"d  a-ng  thosf  thaf  rep 
figures  thJtlshe^nl  be   mV''"''*^^^'^^^''^^  ^^o  tZ^ 

'  ins   18 

,^„    „       ■  '     '     ^ns.  10  o'clock, 

ivy.  How  many  children  have  vArt%1r?     tu  -  .       . 

«„]®wh/""^  T  """""""^  whose>iim  :  their  difference  ■  •  7  ■  <l 

■  Ans.  15  and  6. 

at  6    •tLTrl7ir|31'iog''^l'^'iP^"™'••''■''-"•-'^- 
p.n  is  e,ua,  .  ,  ;Lfo^/rsee!„^  "^^^1:1^^ 

182    P.„.         V  ."       ^'"- **5,000  and  $20,000. 

tio'fi/Zerrri™^-t:i~r;hr7:"r^  '^^  '«■ 

of  the  second  by  the  third  is  Tll;:;:^'  thfCteC ™' 

,S,    TT,„  .  -<"».  4:8::8:  16. 

ISA  Ihe  number  x  exceeds  the  number  «  hv  fho      u  , 
root  of  X,  and  ^  of  ,:  equals  the  A  of  ./III  '•y."'<>  whole 
numbers  with  o*ne  unk,?own  tenn'^      '^     ^A^Ts^nfT" 

oflt.^^a^or'nilrgTbl'ittvl'S,"'"'!^^/^'-''- 
and  my  expense  was  tie  samf  at  h  iftt;!'  tT'' 
mea,,  pr,ce  was  35  shillings.     Required  the 'prtje  rf^a'    ^JS:? 

^'w- 30  and  40  shillings. 
>„l!!i'lr!'r..';™'=.«' '"M  ?  Kv,  my  daughter's  a„,.  will 
to-dav  ? 


""'^  "'   ""'*J'  '■'^  «as  o  years  ago. 


186.  The  .sum  of  two  numbers  is  4 


What  is  her  age 
Ans.  10  years, 
times  their  difference; 


I 


i   B! 


148 


APPENDIX. 


and  the  d.fferenofe  ,i,  of  their  product.    What  .re  the  two  num 

Jn  )^Ia  ,  "  °®*'^'"  ^^'"&  questioned  about  the  number  of  men 

wh    e  sumtTl     Zrr'  '■  '^t"^""'"^^^  ^-  but  3  factors 
wnose  sum  18  31.     What  was  their  number?         Ans  25 

188    Waiter,  your  bill  of  fare  amounts  to  so  much,  does  it 

hat^hl.     K^"   IT''  r'V  T^^^-  ^^^"'  here  is  another  sum 
that  lacks  but  i  dollar  of  the  J  of  the  first,  and  let's  say  no 

IrinnV*'    ^'  '"'  "^^^-^iheless,  hard.  sir.  to  lose  the  double, 
plus  1,  of  the  sq  .arew.of  my  bill.     What  was  the  nmountt 

*  '"  ^w«.  144  dollars. 

ih«t  ft!       T\  °^  '^'^  pandpapa  and  grandson  are  such, 

that  their  nuotient  is  equal  to  |  of  their  product,  and  the  sum 

of  eaeh^  q^o^^ent.aftd. product  is  320.    What  is  the  ^ge 

^^    •  ■   -    ■  Ans.  96  and  3  years.'' 

lyo.   1  mixed  two  pipes'-Af  Vine  ;  one  cost  180  shilJinffs  and 

he  other  140  shillings.     The  first  contained  20  bo  ties  rr^  e 

than  the  second  and  cost  5d.  l^ess  per  bottle.    What  is  the  vZl 

of  a  bottle  of  the  mixture  ?     = ;,.  ^,,.  ^^^-^'^^ 

191.  Ihe  quotient  exceeds  the  divisor  by  half  phis  1  and 
the  sum  of  the  divisor  and  quotient  exceeds  doubeCsoi^ire 
root  of  the  dividend  plus  1.     .R-equired  the  divid!'.  id  visor 


and  quotient. 


Ans.  400,  10,  and  25. 


u!hfr  ^7,"  ^*"^et«J"nni»g  tog\^ttter,  filled  a  basi,,  in  3  hours 
If  the  first  had  run  but  2  hours,  it  would  have  taken  the  second 
6  hours  to  do  the  remainder.  What  time  would  it  take  each 
tunning  alone?  ^...  4 and  12  hours 

193    T^e  father's  age  has  two  factors,  of  which  one  repre- 
sents his  daughter's  age,  and  the  other  is  18  less.     Sqimie  ?h^ 

iXtZVf^^  ''^''  the  daughter's  age,  and  tSis  Lu 
U>  the  sum  of  both  ages,  and  the  general  result  will  be  100 
Required  the  age  of  each.  a  ns.  03  and  2 1 

-,  lit  .'T^e  product  of  two  numbers  exceeds  their  sum  ov  U 
and  the.r  difference  is  2.     What  are  the  two  numbers  ?    ^  ^ 

Ans.  0  and  4, 
_195.   A  number  of  three  figures  is  a  multiple  of  11.  and  rim 
".-iiio  is  quaarupiu  the  huiiuieds.     What  is  the  number? 

Au,.  154. 


> 


he  two  num 

10  and  e. 
Tiber  of  men 
but  3  factors, 

Ans.  25. 
luch,  does  it 
and  receipt 
mother  sum 
let's  say  no 

the  double, 
le  amount  * 
i  dollars. 

n  are  such, 
nd  the  sum 
t  is  the  affe 
3  years. 

liliings  and 
ottles  more 
is  the  value 
.  2s.  5d. 
plus  1,  and 

the  square 
id,  divisor, 

and  25. 

in  3  hours, 
the  second 

t  tal<c  each 

2  hours. 

one  repre- 
Square  this 
this  result 

11  be  100. 
and  21. 

um  oy  14, 

rs? 
)  and  i. 

1 .  and  rlia 

er? 
(6-.  154. 


APPKNUIX 


ue 


at  !hf  ricdit^oH^ft  o1  T  I?^"'"."  ^"\^'  ^^«^  '^»  ^-^  r'«-<i 

t.^7  TT  n^^'"^  ^-^--ratrt  t  etnZrS 
i\es,  the  bird-seller answerpfl  •  If_i_,v.       a  •"i^«-'«>i  lugi- 

■tiv.„u„,her.  which  Tas:t;r''4"aL''rdrY80T'';H 
now  he  reduced  J.  What  was  the  prim  tive  nt  1, 'r  f  w'llil 
B  their  present  number'  ^  I"     w      j  cV"" 

/!«».  I.'jO  mid  84. 

n.nl    ik""!"'"'"'"''  <l™o™ple  3r.<)ther,  and  their  sum  i,  4 

r;h:'trs;r:,"'^''"'■''''''"™^^■^«-•^;"ft"-o:v^^^ 

^  '  ;         -^'tA.  ^5  and  100. 

flute  p,„.  the  square  rL  of^^^lZf  Z  %^.  ''"w^j ^ 
the  price  of  each  instrument,  if  the  sum  of     ,«  tL 
quadruple  the  pounds  given  to  l>oot  bylwrenLr  "'"^  " 

9nn    A  u       ^   ..       .'o  ^^*«.  illo  and  £25. 

diJfs  i.V"""  ri?^'^'''.^^"''"^  ''^  «"«h,  that  the  sun.  <.f  its 
digits  ,s  lb;  and   by  m verting  the  number,  then  addin./tt! 

m."  wtt  ;:r:itrt'"  ^•'"  •^'^  '^''*-^  v'^-- 
of?L,tetroroVits^^^^ 

their  sum  is  to  their  product  •  •  6  •  05  vVKa.  ' '  u'  '  ^ 
ber?  F^wuuci  ..  o  .  .iD.     What  was  the  num- 

202    Two  sisters  have  unequal  .ums  for  their  purchases 

;tn«rfa:^^"™^- -- ^^^- -  f  ?^^*^wi^^ 

^08.  What  IS  ihe  number,  whose  scmaie  reduced   t„   it. 
■luarter,  excneds  by  {  three  times  the  |  ',f  the  nuXV? 

tinferaslrari"iU:.!;r  .\:°T  f ^--<'.  ^^i::  is^ 
squares  of  inarble  of  a^^riainXiTirsi^i^St'Z uWt/ 
n.e  mason  answered,  that  if  the  length  was  but  double  Te 

l.S* 


150 


APPENDIX. 


CJD 


breadth,  it  would  have  taken   800  less.     What  do  vou 
dudo  fi-oni  this  answer  ?  '  i  uo  ^ou 

Ans.  That  it  would  have  taken  4000. 

.Jf'  ^  T^'^^"^  g^^«  2^  "^'  his  profit  to  the  poor.  At  thr 
>ear  s  end  his  alms  amounted  to  $390.  I  demand  what  vvas 
the  amount  of  his  sales,  if  half  was  at  10,  I  at  15  and  k 
remainder  at  18  per  cent,  profit  ?  ^Ans.  $6(^000 

fhnT' J^^  '"""  ""f  the'foJr  terms  of  a  proposition  is  63- 
the  first  ,s  4  more  than  thVk:c.ond ;  the  quotient  of  the  th  rd 
by  the  second  ,s>8i;  and  .the  product  of  the  means  I  136 
Required  the  four  terms.    ' .;: . .  Ans.  8  :  4  •  •  34     17 

^  A*jf .  *  :  o  : ;  4  •  1  'i . 


TTDC   IWD. 


do 


you  cjD 


ikeii  4000. 

oor.     At  the 
nd  what  was 
15,  and  the 
.  160,000. 

ber  was  rep 
representing 
at  the  sum 
t  what  num. 
No.  210. 

ition  is  63 ; 
of  the  third 
sans  i,«  136. 
:  34  :  17. 

^ion  i'i  576 ; 
term  k  10, 
What  an> 
:4-  1','. 


I 


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tun  in  Puor  Sc/^oU 


cnuls■^lA^  uw^muiiis'  fikst 

•in.-"  PRAfTJi  VI  ..   -"^-  Lie 

*  ■'  u  ','        ''"«■  Dioct-se  of  Boston.  ",'!' 

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